\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 149, pp. 1--20.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/149\hfil Exact behavior of singular solutions] {Exact behavior of singular solutions to Protter's problem with lower order terms} \author[A. Nikolov, N. Popivanov\hfil EJDE-2012/149\hfilneg] {Aleksey Nikolov, Nedyu Popivanov} % in alphabetical order \address{Aleksey Nikolov \newline Faculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria} \email{lio6kata@yahoo.com} \address{Nedyu Popivanov \newline Faculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria} \email{nedyu@fmi.uni-sofia.bg} \thanks{Submitted May 8, 2012. Published August 29, 2012.} \subjclass[2000]{35L05, 35L20, 35D05, 35A20} \keywords{Wave equation; boundary value problems; generalized solutions; \hfill\break\indent singular solutions; propagation of singularities} \begin{abstract} For the (2+1)-D wave equation Protter formulated (1952) some boundary value problems which are three-dimensional analogues of the Darboux problems on the plane. Protter studied these problems in a 3-D domain, bounded by two characteristic cones and by a planar region. Now it is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions, because of the strong power-type singularity which appears in the generalized solution. In the present paper we consider the wave equation involving lower order terms and obtain new a priori estimates describing the exact behavior of singular solutions of the third boundary value problem. According to the new estimates their singularity is of the same order as in case of the wave equation without lower order terms. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} We denote points in $\mathbb{R}^3$ by $(x,t)=(x_1,x_2,t)$ and consider the wave equation involving lower order terms $$Lu\equiv u_{x_1x_1}+u_{x_2x_2}-u_{tt}+b_1u_{x_1}+b_2u_{x_2}+bu_{t}+cu=f \label{eq0p1}$$ in a simply connected region $\Omega_0:=\{(x,t):00. \end{theorem} In the same paper one can find a proof of the uniqueness of the treated problem. Note that the generalized solutions in this theorem have singularities at the vertex O of the cone \Sigma_{2,0} and that these singularities do not propagate in the direction of the bicharacteristics on the characteristic cone \Sigma_{2,0}. For results concerning the propagation of singularities for solutions of second order operators see H\"{o}rmander \cite[Chapter 24.5]{H}. On the other hand, Hristov, Popivanov and Schneider in \cite{last} (see Theorem 4.4 there in) obtained some upper bounds for all the solutions of this problem, considering the case that the coefficients b_1, b_2, b, c and \alpha are smooth functions in \bar{\Omega}_0 (the coefficients of the equation \eqref{eq0p1} in polar coordinates, like it is in Theorem \ref{thmG}, do not depend on \varphi) and also assuming the function f \in C(\bar{\Omega}_0) to be of the form $$f(\varrho ,\varphi ,t)=f_{n}^{(1)}(\varrho ,t)\cos n\varphi +f_{n}^{(2)}(\varrho ,t)\sin n\varphi, n\in\mathbb{N}. \label{eqf}$$ These upper bounds can be written of the form: $$|u(x,t)|\leq C_0\max_{\bar{\Omega}_0}{\{|f_{n}^{(1)}|+|f_{n}^{(2)}|\}} |x|^{-n-\psi(K)}, \label{eqest1}$$ where C_0 is a positive constant, \[ K:=\max\big\{\sup_{\bar{\Omega}_0} |b_1|, \sup_{\bar{\Omega}_0} |b_2|, \sup_{\bar{\Omega}_0} |b|, \sup_{\bar{\Omega}_0} |c|, \sup_{0\leq |x| \leq 1} |\alpha(|x|)|\big\}$ and $\psi(K)$ is a positive function which blows up as $K$ blows up. In the present paper this estimate is improved by the following main result \begin{theorem} \label{mainres} Let the right-hand side function $f$ in the equation \eqref{eq0p1} is of the form \eqref{eqf}, $b_1, b_2, b, c\in C(\bar{\Omega}_0)$, $\alpha\in C^1([0,1])$, $f_n^{(i)}\in C(\bar{\Omega}_0)$, $i=1,2$ and $a_1,a_2,b,c$ are functions of $(|x|,t)$, $\alpha=\alpha(|x|)$, where $a_1:=b_1\cos(\arctan \frac{x_2}{x_1})+b_2\sin(\arctan \frac{x_2}{x_1})$, $a_2:=|x|^{-1}(b_2\cos (\arctan \frac{x_2}{x_1})-b_1\sin(\arctan \frac{x_2}{x_1}))$. Then for the generalized solution $u(x,t)$ of Problem $\mathbf{P}_\alpha$ the following estimate $$|u(x,t)|\leq C_\sigma\max_{\bar{\Omega}_0}{\{|f_{n}^{(1)}|+|f_{n}^{(2)}|\}} |x|^{-n-\sigma} \label{eqest2}$$ holds, where $\sigma$ is an arbitrary positive number and $C_\sigma$ is a positive constant depending on $\sigma$, $n$ and all coefficients of \eqref{eq0p1}. \end{theorem} \begin{remark} \rm A new point here, as distinct from \eqref{eqest1}, is the fact that the order of singularity does not depend on the lower order terms of \eqref{eq0p1} and on the boundary coefficient $\alpha$. \end{remark} Comparing this estimate with the lower bound of the singular solutions found in Theorem \ref{thmG}, we see that we have obtained their exact asymptotic behavior. First, in this work we follow the exposition of Hristov et al \cite{last} until Theorem 4.4. This takes the next three sections. In Section 2 Problem $P_{\alpha }$ is reduced to a two-dimensional problem in the following steps. First, we transform equation \eqref{eq0p1} in polar coordinates, i.e. $$Lu=\frac{1}{\varrho }(\varrho u_{\varrho })_{\varrho }+\frac{1}{\varrho ^2} u_{\varphi \varphi }-u_{tt}+a_1u_{\varrho }+a_2u_{\varphi }+bu_{t}+cu=f, \label{eq0polar}$$ ($a_1:=b_1cos \varphi+b_2sin \varphi$, $a_2:=\varrho^{-1}(b_2cos \varphi-b_1sin \varphi)$), considering, as noted before, a polar symmetry of $a_1,a_2,b,c$ and $\alpha$, and a special form of the right-hand side \eqref{eqf}. Next, we ask for generalized solution of the form $$u(\varrho ,\varphi ,t)=u_{n}^{(1)}(\varrho ,t)\cos n\varphi +u_{n}^{(2)}(\varrho ,t)\sin n\varphi. \label{equ}$$ Thus separating the variables we succeed in reducing the problem to a two-dimen\-sional one for functions $\{u_n^{(1)}(\varrho,t),u_n^{(2)}( \varrho,t)\}$, called Problem $P_{\alpha,1}$. Finally, using characteristic coordinates $\xi=1-\varrho-t$, $\eta=1-\varrho+t$ and new functions $$u_n^{(i)}(\xi,\eta):=z_n^{(i)}(\varrho,t):=\varrho^{\frac{1}{2} }u_n^{(i)}(\varrho,t), i=1,2, \label{eq0pnewf}$$ we obtain a system for $\{u_n^{(1)}(\xi,\eta),u_n^{(2)}(\xi,\eta)\}$, called Problem $P_{\alpha,2}$. In Section 3 an equivalent integral equation system of Problem $P_{\alpha,2}$ is constructed. In Section 4 are presented some results from \cite{last} which we use in the next section. Also, here is formulated the main result of \cite{last}, Theorem 4.4, which ensures the existence of a generalized solution of the two-dimensional Problem $P_{\alpha,2}$ and gives upper bounds of possible singularity. Using this theorem, after the inverse transformation to Problem $P_{\alpha}$, one comes to \eqref{eqest1}. In Section 5 we prove Theorem \ref{mainres}, the main result of this work. The next Section 6 is dedicated to the singular solutions. Modifying a little the proof of Theorem \ref{thmG}, we deduce the following result. \begin{theorem} \label{thmGN} Let $\alpha\geq 0;$ $b_1$, $b_2$, $b$, $c\in C^{1}({\bar{\Omega}}_0\backslash O)$ and $$b_1=a_1(|x|,t)\cos(\arctan x_2/x_1),\quad b_2=a_1(|x|,t)\sin(\arctan x_2/x_1)$$ with some function $a_1(|x|,t)$ for which $a_1\geq|b|, a_1\geq2|x| c$. Then for each function of the form \begin{gather*} f(x,t)=f_n(|x|,t)\cos n(\arctan x_2/x_1)\quad \text{or}\\ f(x,t)=f_n(|x|,t)\sin n(\arctan x_2/x_1), \quad n\in\mathbb{N} \end{gather*} in the right-hand side of the equation, satisfying the following conditions: $f_n\in C(\bar{\Omega}_0),\quad f_n\not\equiv0\quad\text{in }\Omega_0 ,\quad \text{either }f_n\geq0\text{ or }f_n\leq0\quad\text{in }\Omega_0,$ the corresponding generalized solution $u_{n}$ of the problem $P_{\alpha }$ satisfies the estimate $$\label{qram2} |u_{n}(x,t)|\geq C_0|x|^{-n}|\cos n(\arctan \frac{x_2}{x_1})|,\quad C_0=\text{const}>0$$ in some neighborhood of $O(0,0,0)$. \end{theorem} The difference between this theorem and Theorem \ref{thmG} is that we have the same result for a wider class of right-hand side functions and, as well, in \eqref{qram2} we estimate $|u_{n}(x,t)|$, while in \eqref{qram} is estimated the restriction $|u_{n}(x,t)|_{t=|x|}$. In the case of wave equation without lower order terms and $\alpha\equiv0$, Theorem \ref{thmGN} is in correspondence with the results deduced so far. Actually, in \cite{DPP} one can find an asymptotic expansion of the generalized solution at the origin. According to this work, the order of singularity of the solution is less than $n$ only if some orthogonality conditions are fulfilled, namely if the function $f_n$ is orthogonal to some solutions of the adjoint homogeneous problem $P2^{\ast}$. If $f_n$ does not change its sign, a necessary orthogonality condition is not fulfilled. In the case of wave equation with lower order terms, we do not know such orthogonality conditions controlling'' the order of singularity of the corresponding solution. \subsection*{Open Question 2} Can one find some orthogonality conditions in the case of the equation \eqref{eq0p1}, under which we have a lower order of singularity? \section{Preliminaries} As we noted in the previous section, we consider \eqref{eq0p1} in polar coordinates (see \eqref{eq0polar}) in case that the right-hand side of the equation is of the form \eqref{eqf} and we ask for the generalized solution to be of the form \eqref{equ}. Here we assume that all coefficients of \eqref{eq0polar} depend only on $\varrho$ and $t$, and we set $\alpha(x)\equiv\alpha(\varrho)\in C^1[0,1]$. Thus from \eqref{eq0p1} we obtain the system $$\label{eq2p1} \begin{gathered} \frac{1}{\varrho }(\varrho u_{n,\varrho }^{(1)})_{\varrho }-u_{n,tt}^{(1)}+a_1u_{n,\varrho }^{(1)}+bu_{n,t}^{(1)}+(c-\frac{n^2} {\varrho ^2})u_{n}^{(1)}+na_2u_{n}^{(2)}=f_{n}^{(1)}, \\ \frac{1}{\varrho }(\varrho u_{n,\varrho }^{(2)})_{\varrho }-u_{n,tt}^{(2)}+a_1u_{n,\varrho }^{(2)}+bu_{n,t}^{(2)}+(c-\frac{n^2} {\varrho ^2})u_{n}^{(2)}-na_2u_{n}^{(1)}=f_{n}^{(2)}. \end{gathered}$$ To deal with singularities on $t=\varrho$, especially at $(0,0)$, we consider \eqref{eq2p1} in the domain $G_{\varepsilon }=\{(\varrho ,t):t>0,\varepsilon +t<\varrho <1-t\},\varepsilon>0$ which is bounded by the disc $S_0 =\{(\varrho ,t):t=0,0<\varrho <1\}$, and $S_1=\{(\varrho ,t):\varrho =1-t\},\quad S_{2,\varepsilon }=\{(\varrho ,t):\varrho =t+\varepsilon \}$ and treat the following problem (omitted the index $n$): \subsection*{Problem $\boldsymbol{P_{\alpha ,1}}$} Find solutions $u=(u^{(1)},u^{(2)})$ of system \eqref{eq2p1} which satisfy $u^{(i)}|_{S_1\cap\partial G_\varepsilon}=0,\quad [u_t^{(i)}+\alpha(\varrho)u^{(i)}]|_{S_0\cap\partial G_\varepsilon}=0,\quad i=1,2.$ \begin{definition} \rm A function $u=(u^{(1)},u^{(2)})(\varrho ,t)$ is called a \textsl{generalized solution} of Problem $P_{\alpha ,1}$ in $G_{\varepsilon }$, $\varepsilon >0$, if: \begin{enumerate} \item $u\in C^{1}({\bar{G}}_{\varepsilon })$, $[u_{t}^{(i)}+\alpha (\varrho )u^{(i)}]\bigr\rvert_{S_0\cap \partial G_{\varepsilon }}=0$, $u^{(i)} \bigr\rvert_{S_1\cap \partial G_{\varepsilon }}=0,i=1,2;$ \item The equalities \begin{align*} &\int_{G_{\varepsilon }}\big[u_{t}^{(1)}v_{1,t}-u_{\varrho }^{(1)}v_{1,\varrho }+\big( a_1u_{\varrho }^{(1)}+bu_{t}^{(1)}+(c -\frac{n^2}{\varrho ^2})u^{(1)}+na_2u^{(2)} -f^{(1)}\big) v_1\big] \varrho d\varrho \,dt \\ &=\int_{S_0\cap \partial G_{\varepsilon }}\alpha (\varrho )u^{(1)}v_1\varrho \,d\varrho ,\\ &\int_{G_{\varepsilon }}[u_{t}^{(2)}v_{2,t}-u_{\varrho }^{(2)}v_{2,\varrho }+\big( a_1u_{\rho }^{(2)}+bu_{t}^{(2)} +(c-\frac{n^2}{\varrho ^2})u^{(2)}-na_2u^{(1)} -f^{(2)}\big) v_2\big] \varrho d\varrho \,dt\\ &=\int_{S_0\cap \partial G_{\varepsilon }}\alpha (\varrho )u^{(2)}v_2\varrho \,d\varrho \end{align*} hold for all $v_1,v_2\in V_{\varepsilon }^{(1)}=\{{v\in C^{1}({\bar{G}}_{\varepsilon }):[v_{t}+(\alpha +b)v]\bigr\rvert_{S_0\cap \partial G_{\varepsilon }}=0,v \bigr\rvert_{S_{2,\varepsilon }\cap \partial G_{\varepsilon }}=0\}.}$ \end{enumerate} \label{def2.1} \end{definition} Introducing a new function $$z^{(i)}(\varrho ,t)=\varrho ^{\frac{1}{2}}u^{(i)}(\varrho ,t) =z^{(i)}(\varrho(\xi,\eta) ,t(\xi,\eta))=:U^{(i)}(\xi,\eta),\ i=1,2, \label{eqU}$$ in characteristic coordinates $$\label{charcoord} \xi =1-\varrho -t,\quad \eta =1-\varrho +t$$ we obtain the system $$\begin{gathered} U_{\xi \eta }^{(1)}-A_1U_{\xi }^{(1)}-B_1U_{\eta }^{(1)}-C_1U^{(1)}-D_1U^{(2)}=F^{1}(\xi ,\eta )\text{ \quad in }D_{\varepsilon }, \\ U_{\xi \eta }^{(2)}-A_2U_{\xi }^{(2)}-B_2U_{\eta }^{(2)}-C_2U^{(2)}-D_2U^{(1)}=F^2(\xi ,\eta )\text{ \quad in }D_{\varepsilon }, \end{gathered} \label{eqdarbu}$$ where $D_{\varepsilon }=\{(\xi ,\eta ):0<\xi <\eta <1-\varepsilon \}$ and $$F^{(i)}(\xi ,\eta )=\frac{1}{4\sqrt{2}}(2-\xi-\eta )^{\frac{1}{2} }f^{(i)}(\varrho (\xi ,\eta ),t(\xi ,\eta )), i=1,2, \label{eqF}$$ $$\begin{gathered} A_1=A_2=\frac{1}{4}(a_1+b), \quad B_1=B_2=\frac{1}{4} (a_1-b), \\ D_2=-D_1=\frac{1}{4}na_2,\quad C_1=C_2=\frac{1}{4}\Big\{ \frac{4n^2-1}{(2-\xi -\eta )^2}+\frac{a_1}{2-\xi -\eta }-c\Big\} . \end{gathered} \label{eqABCD}$$ Note, that Problem $P_{\alpha,1}$ is reduced to the Darboux-Goursat problem for the system \eqref{eqdarbu} in $D_\varepsilon$. Note also, that if we consider this problem in $D_0$ , then the coefficients $C_i, D_i (i=1,2)$ are singular at the point $(1,1)$. To investigate the smoothness or the singularities of solutions at the original problem $P_{\alpha }$ on $\Sigma_{2,0}$, we are looking for classical solutions for the system \eqref{eqdarbu} not only in the domain $D_{\varepsilon }$, but also in the domain $D_{\varepsilon }^{(1)}:=\{(\xi ,\eta ):0<\xi <\eta <1,0<\xi <1-\varepsilon \},\quad \varepsilon >0,$ where $D_{\varepsilon }\subset D_{\varepsilon }^{(1)}$. Thus we come to the following question. \subsection*{Problem $\boldsymbol{P_{\alpha ,2}}$} Find solutions $(U^{(1)},U^{(2)})(\xi,\eta)$ of system \eqref{eqdarbu} in $D_{\varepsilon }^{(1)}$, which satisfy the boundary conditions $$U^{(i)}(0,\eta )=0,(U_{\eta }^{(i)}-U_{\xi }^{(i)})(\xi ,\xi )+\alpha (1-\xi )U^{(i)}(\xi ,\xi )=0, \label{bound}$$ $i=1,2$, $\xi \in (0,1-\varepsilon )$, $\eta\in(0,1)$. \section{A system of integral equations for problem $\boldsymbol{P_{\alpha,2}}$} We consider a point $(\xi_0,\eta_0)\in D_\varepsilon^{(1)}$ and rectangle $R$, triangle $T$ defined by \begin{gather*} R:=\{(\xi ,\eta ):0<\xi <\xi_0,\xi_0<\eta <\eta_0\}, \\ T:=\{(\xi ,\eta ):0<\xi <\xi_0,\xi <\eta <\xi_0\}. \end{gather*} By use of Green's theorem in $$\begin{gathered} I_R^{(i)}:=\iint_{R}U_{\xi \eta }^{(i)}(\xi ,\eta )\,d\xi d\eta \,=\int_0^{\xi_0} \Big(\int_{\xi_0}^{\eta_0}U_{\xi \eta }^{(i)}(\xi ,\eta )\,d\eta \Big)\,d\xi, \\ I_T^{(i)}:=\iint_{T}U_{\xi \eta }^{(i)}(\xi ,\eta )\,d\xi d\eta \,=\int_0^{\xi_0}\Big(\int_{\xi }^{\xi_0}U_{\xi \eta }^{(i)}(\xi ,\eta )\,d\eta \Big)\,d\xi , \end{gathered} \label{eqIRT}$$ $i=1,2$, and the boundary conditions \eqref{bound} we obtain $$I_R^{(i)}+2I_T^{(i)}=U^{(i)}(\xi_0,\eta_0) -\int_0^{\xi _0}\alpha (1-\xi)U^{(i)}(\xi ,\xi )\,d\xi . \label{eqIR+2T}$$ We set $p^{(i)}:=U_{\xi }^{(i)}$, $q^{(i)}:=U_{\eta }^{(i)}$ and define (see \eqref{eqdarbu}) $$\begin{gathered} E^{(1)}(\xi, \eta):=[F^1+A_1p^{(1)}+B_1q^{(1)} +C_1U^{(1)}+D_1U^{(2)}](\xi ,\eta ), \\ E^{(2)}(\xi, \eta):=[F^2+A_2p^{(2)}+B_2q^{(2)} +C_2U^{(2)}+D_2U^{(1)}](\xi ,\eta ). \end{gathered} \label{eqE}$$ Using \eqref{eqIRT} - \eqref{eqE} and \eqref{eqdarbu} we obtain six integral equations ($i=1,2$) \begin{aligned} U^{(i)}(\xi_0,\eta_0) &=\int_0^{\xi_0}\Big(\int_{\xi_0}^{\eta_0}E^{(i)}(\xi ,\eta )\,d\eta \Big)d\xi+2\int_0^{\xi_0}\Big(\int_0^\eta E^{(i)}(\xi ,\eta )\,d\xi \Big)d\eta\\ &\quad +\int_0^{\xi_0}\alpha(1-\xi)U^{(i)}(\xi,\xi)d\xi, \label{IU} \end{aligned} $$p^{(i)}(\xi_0,\eta_0) =\int_0^{\xi_0}E^{(i)}(\xi ,\xi_0 )d\xi+\int_{\xi_0}^{\eta_0}E^{(i)}(\xi_0,\eta)d\eta +\alpha(1-\xi_0)U^{(i)}(\xi_0,\xi_0), \label{Ip}$$ $$q^{(i)}(\xi_0,\eta_0)=\int_0^{\xi_0}E^{(i)}(\xi ,\eta_0 )d\xi \label{Iq}$$ The system \eqref{IU}--\eqref{Iq} is equivalent to the system \eqref{eqdarbu} with the boundary conditions \eqref{bound}. \begin{remark}\label{rmk3.1} \rm We recall that in Section 2 the index $n$ in system \eqref{eq2p1} was omitted. We see that in \eqref{eqdarbu} the coefficients $C_i,D_i$ $(i=1,2)$ depend on $n$, where on the right-hand side we have $F^{(i)}(\xi ,\eta )=\frac{1}{4\sqrt{2}}(2-\xi -\eta )^{\frac{1 }{2}}f_n^{(i)}(\varrho (\xi ,\eta ),t(\xi ,\eta )).$ Therefore for fixed $n\in\mathbb{N}$ solutions $(U^{(1)},U^{(2)})$ of the integral equation system \eqref{IU} - \eqref{Iq} depend on $n$ and will be later marked by $(U_n^{(1)},U_n^{(2)})$, which gives functions $(u_n^{(1)},u_n^{(2)})$ by relation $\varrho ^{\frac{1}{2}}u^{(i)}(\varrho ,t)=U_n^{(i)}(\xi,\eta)$ (see \eqref{eqU}). Furthermore we observe that classical solutions $(U_n^{(1)},U_n^{(2)})\in C^1(\bar{D}_\varepsilon^{(1)})$, $U_{n,\xi\eta}^{(i)}\in C(\bar{D}_\varepsilon^{(1)})$ of the integral equation system define functions $(u_n^{(1)},u_n^{(2)})$ which are generalized solutions of Problem $P_{\alpha,1}$ in $\bar{G}_0\backslash(0,0)$. \end{remark} \section{Solutions of the system and first upper estimates} We define in $D_\varepsilon^{(1)}$ functions $(U_m^{(i)},p_m^{(i)},q_m^{(i)})$, $i=1,2$, $m\in\mathbb{N}$, by the formulas \begin{aligned} U_{m+1}^{(i)}(\xi_0,\eta_0) &=\int_0^{\xi_0}\Big(\int_{\xi _0}^{\eta_0}E_{m}^{(i)}(\xi ,\eta ) \,d\eta \Big)\,d\xi +2\int_0^{\xi_0}\Big(\int_0^{\eta }E_{m}^{(i)}(\xi ,\eta ) \,d\xi \Big)\,d\eta \\ &\quad +\int_0^{\xi_0}\alpha (1-\xi )U_{m}^{(i)}(\xi ,\xi )\,d\xi ,\quad \ i=1,2;\quad m=0,1,2\dots \\ p_{m+1}^{(i)}(\xi_0,\eta_0) =&\int_0^{\xi_0}E_{m}^{(i)}(\xi ,\xi_0)\,d\xi +\int_{\xi_0}^{\eta _0}E_{m}^{(i)}(\xi_0,\eta )\,d\eta \\ &\quad +\alpha (1-\xi_0)U_{m}^{(i)}(\xi_0,\xi_0),\ \quad i=1,2;\quad m=0,1,2\dots \\ q_{m+1}^{(i)}(\xi_0,\eta_0) &=\int_0^{\xi _0}E_{m}^{(i)}(\xi ,\eta_0)\,d\xi ,\quad i=1,2;\quad m=0,1,2\dots \\ U_0^{(i)}(\xi_0,\eta_0)&=0,\quad p_0^{(i)}(\xi_0,\eta_0)=0, q_0^{(i)}(\xi_0,\eta_0)=0,\quad i=1,2, \end{aligned} \label{eqappr} in $D_\varepsilon^{(1)}$, where $$\begin{gathered}\label{eqEm} E_{m}^{(1)}(\xi ,\eta ) :=[F^{1}+A_1p_{m}^{(1)}+B_1q_{m}^{(1)}+C_1U_{m}^{(1)}+D_1U_{m}^{(2)}](\xi ,\eta ), \\ E_{m}^{(2)}(\xi ,\eta ) :=[F^2+A_2p_{m}^{(2)}+B_2q_{m}^{(2)}+C_2U_{m}^{(2)}+D_2U_{m}^{(1)}](\xi ,\eta ). \end{gathered}$$ Now we formulate some results from Hristov et al \cite{last} which we use later. \begin{lemma}[\cite{last}] \label{lemma41} Let for $(\xi_0,\eta_0)\in D_{\varepsilon }^{(1)}=\{(\xi ,\eta ):0<\xi <\eta <1,0<\xi <1-\varepsilon \},\varepsilon >0$, and $\mu\in\mathbb{R}_+$ define $I_\mu:=\int_0^{\xi_0}\Big(\int_{\xi_0}^{\eta_0}(2-\xi-\eta)^{-\mu-2} d\eta \Big)d\xi+2\int_0^{\xi_0}\Big(\int_\xi^{\xi_0}(2-\xi-\eta)^{-\mu-2} d\eta \Big)d\xi.$ Then $I_\mu\leq\frac{1}{\mu(\mu+1)}(2-\xi_0-\eta_0)^{-\mu}.$ \end{lemma} As we mentioned in the introduction, we treat in this paper the equation \eqref{eq0p1} in case that its coefficients are continuous in $\bar{\Omega}_0$, so we may set $$\label{Cd1} \sup_{\bar{\Omega}_0}\{|b_1|,|b_2|,|b|\}\leq K_1, \quad \sup_{\bar{\Omega}_0}|c|\leq K_0,\quad \sup_{[0,1]}|\alpha(\varrho)|\leq K_\alpha.$$ Then, from \eqref{eqABCD} we obtain the following bounds \begin{gather*} |a_1|\leq2K_1, |a_2|\leq\frac{2K_1}{\rho}, |A_1|=|A_2|\leq\frac{3K_1}{4}, \\ |B_1|=|B_2|\leq\frac{3K_1}{4}, |D_1|=|D_2|\leq\frac{nK_1}{2\rho}=\frac{nK_1 }{2-\xi-\eta}, \\ |C_1|=|C_2|\leq\frac{\nu(\nu+1)}{(2-\xi-\eta)^2}+\frac{K_1}{2(2-\xi-\eta)}+ \frac{K_0}{4}, \end{gather*} where $\nu:=n-\frac{1}{2}$. According to \eqref{eqEm} $E_{m}^{(i)}(\xi ,\eta ):=[F^i+A_ip_{m}^{(i)}+B_iq_{m}^{(i)}+C_iU_{m}^{(i)}+D_iU_{m}^{( \gamma _i)}](\xi ,\eta ),$ with $\gamma _{1}=2,\gamma _{2}=1$ and thus for $i=1,2$ we have \label{eqestEm} \begin{aligned} &|(E_{m}^{(i)}-E_{m-1}^{(i)})(\xi ,\eta )|\\ &\leq \big\{ \frac{\nu (\nu +1)}{ (2-\xi -\eta )^{2}}+\frac{K_{1}}{2(2-\xi -\eta )}+\frac{K_0}{4}\big\} |U_{m}^{(i)}-U_{m-1}^{(i)}| \\ &\quad +\frac{(\nu +1/2)K_{1}}{2-\xi -\eta }|U_{m}^{(\gamma _i)}-U_{m-1}^{(\gamma _i)}|+\frac{3K_{1}}{4}|p_{m}^{(i)}-p_{m-1}^{(i)}|+\frac{3K_{1}}{4} |q_{m}^{(i)}-q_{m-1}^{(i)}|. \end{aligned} \begin{lemma}[\cite{last}] \label{lemma42} Let the conditions \eqref{Cd1} be fulfilled and there exists a constant $A>0$, such that \begin{gather*} |(U_m^{(i)}-U_{m-1}^{(i)})(\xi_0, \eta_0)|\leq A(2-\xi_0-\eta_0)^{-\mu},\\ |(p_m^{(i)}-p_{m-1}^{(i)})(\xi_0, \eta_0)|\leq\mu A(2-\xi_0-\eta_0)^{-\mu-1},\\ |(q_m^{(i)}-q_{m-1}^{(i)})(\xi_0, \eta_0)|\leq\mu A(2-\xi_0-\eta_0)^{-\mu-1}, \end{gather*} where $\mu\in\mathbb{R}_+, \mu>\nu=n-1/2, m\in\mathbb{N}$. If the parameter $\delta_\nu$ is such, that $$\label{deltanu} (\mu-\nu)(\mu+\nu+1)\geq\delta_\nu\mu(\mu+1)+(3\mu+2\nu+2)K_1 +2(\mu+1)K_\alpha+K_0,$$ then for $m\in\mathbb{N}$, $i=1,2$ we have \begin{gather*} |(U_{m+1}^{(i)}-U_m^{(i)})(\xi_0, \eta_0)|\leq A(1-\delta_\nu)(2-\xi_0-\eta_0)^{-\mu},\\ |(p_{m+1}^{(i)}-p_m^{(i)})(\xi_0, \eta_0)|\leq\mu A(1-\delta_\nu)(2-\xi_0-\eta_0)^{-\mu-1},\\ |(q_{m+1}^{(i)}-q_m^{(i)})(\xi_0, \eta_0)|\leq\mu A(1-\delta_\nu)(2-\xi_0-\eta_0)^{-\mu-1}. \end{gather*} \end{lemma} \begin{lemma}[\cite{last}] \label{lemma43} Let now $\nu=n-1/2,n\in\mathbb{N}$ be fixed. If the parameter $\mu$ is large enough, $\mu>\nu$, then $$\label{largemu} (\mu-\nu)(\mu+\nu+1)-[(3\mu+2\nu+2)K_1 +2(\mu+1)K_\alpha+K_0]>0$$ and we can choose the parameter $\delta_\nu>0$, such that the condition \eqref{deltanu} to be fulfilled. \end{lemma} In \cite{last} the integral equation system \eqref{IU}--\eqref{Iq} is solved by the successive approximations method and the following important theorem is proved. \begin{theorem}[\cite{last}] \label{thm44} Let $n\in\mathbb{N}$ be fixed. Assume: \begin{itemize} \item[(i)] $a_1=b_1cos \varphi+b_2sin \varphi, a_2=\varrho^{-1}(b_2cos \varphi-b_1sin \varphi), b, c$ are functions of $(\varrho,t)$, $\alpha=\alpha(\rho)$; \item[(ii)] $b_1, b_2, b, c\in C(\bar{\Omega}_0)$, $\alpha(\varrho)\in C^1([0,1])$, $f_n^{(i)}\in C(\bar{\Omega}_0)$, $i=1,2$; \item[(iii)] the parameter $\mu=\mu_n$ is such large, that $(\mu-\nu)(\mu+\nu+1)>(3\mu+2\nu+2)K_1+2(\mu+1)K_\alpha+K_0$ (see Lemma \ref{lemma43}). \end{itemize} Then there exists a classical solution $(U_n^{(1)},U_n^{(2)})\in C^1(\bar{D}_\varepsilon^{(1)})$, $U_{n,\xi_0\eta_0}^{(i)}\in C(\bar{D}_\varepsilon^{(1)})$ of Problem $P_{\alpha,2}$ and the following estimates hold: $$\label{estimateU} \begin{gathered} |U_n^{(i)}(\xi,\eta)|\leq A_\mu\delta_\nu^{-1}(2-\xi-\eta)^{-\mu},\\ |U_{n,\xi}^{(i)}(\xi,\eta)|\leq\mu A_\mu\delta_\nu^{-1}(2-\xi-\eta)^{-\mu-1},\\ |U_{n,\eta}^{(i)}(\xi,\eta)|\leq\mu A_\mu\delta_\nu^{-1}(2-\xi-\eta)^{-\mu-1}, \end{gathered}$$ where \begin{gather*} A_\mu:=\frac{1}{\mu(\mu+1)}\max_{\bar{G}_0}| \frac{1}{4\sqrt{2}}(2\varrho)^{\mu+\frac{5}{2}}f_n^{(i)}(\varrho,t)|, \\ \delta_\nu:=\frac{1}{\mu(\mu+1)}\{(\mu-\nu)(\mu+\nu+1)-[(3\mu+2\nu+2)K_1 +2(\mu+1)K_\alpha+K_0]\} \end{gather*} \end{theorem} After the inverse transformation to Problem $P_\alpha$ (using the relation \eqref{eqU}), we see that the first estimate of \eqref{estimateU} is equivalent to \eqref{eqest1}. Next, we aim to refine this result. \section{New (exact) upper estimates} \begin{theorem}\label{thmnew} Let $n\in\mathbb{N}$ be fixed and the conditions (i) and (ii) from Theorem \ref{thm44} be fulfilled. Then for each number $\sigma>0$ there exists a positive constant $C_\sigma$, such that the inequalities $$\label{newestimates} \begin{gathered} |U_n^{(i)}(\xi,\eta)|\leq C_\sigma\max_{\bar{D}_0^{(1)}} |F^{(i)}|(2-\xi-\eta)^{-\nu-\sigma},\\ |U_{n,\xi}^{(i)}(\xi,\eta)|\leq(\nu+\sigma)C_\sigma\max_{\bar{D}_0^{(1)}} |F^{(i)}|(2-\xi-\eta)^{-\nu-\sigma-1},\\ |U_{n,\eta}^{(i)}(\xi,\eta)|\leq(\nu+\sigma)C_\sigma\max_{\bar{D}_0^{(1)}} |F^{(i)}|(2-\xi-\eta)^{-\nu-\sigma-1} \end{gathered}$$ hold in $\bar{D}_\varepsilon^{(1)}$, $i=1,2$. $C_\sigma>0$ depends on the numbers $\nu,\sigma,K_1,K_0$ and $K_\alpha$. \end{theorem} \begin{proof} Let us choose and fix some $\mu >\nu$ satisfying Lemma \ref{lemma43}. Next, we choose and fix an arbitrary positive number $\sigma$, such that $\nu +\sigma <\mu$. Further, we choose $\delta_\nu\in (0,1)$ satisfying the condition \eqref{deltanu} from Lemma \ref{lemma42}. From Lemma \ref{lemma43} we see that it is possible. Now we introduce the positive number $$\label{tau} \tau :=\max\{(1-\delta_\nu),\theta \}<1,$$ where $\theta :=\frac{\nu (\nu +1)}{(\nu +\sigma /2)(\nu +\sigma /2+1)}.$ For shortness in the further calculations, we denote $N(K_{1},K_0,K_{\alpha }):=\frac{(5\nu +3\sigma +2)K_1+K_0+2(\nu +\sigma+1/2)K_\alpha+1} {(\nu +\sigma-1/2)(\nu +\sigma+1/2)}.$ Note that $N(K_{1},K_0,K_{\alpha })>0$ and $\frac{\nu (\nu +1)}{(\nu +\sigma )(\nu +\sigma +1)}<\theta.$ Next, we divide $D_\varepsilon^{(1)}$ by the line $2-\xi-\eta=\frac{1}{N(K_1, K_0, K_\alpha)^2}\left(\theta- \frac{\nu(\nu+1)}{ (\nu+\sigma)(\nu+\sigma+1)}\right)^2$ and obtain two parts: \begin{align*} D1&:=\Big\{(\xi,\eta):0<\xi<\eta<1,\phantom{\frac{1}{2}} 0<\xi<1-\varepsilon, \\ &\quad (2-\xi-\eta)^{1/2}>\frac{1}{N(K_1, K_0, K_\alpha)}\Big(\theta- \frac{ \nu(\nu+1)}{(\nu+\sigma)(\nu+\sigma+1)}\Big)\Big\}, \end{align*} and \begin{align*} D2&:=\Big\{(\xi,\eta):0<\xi<\eta<1,\phantom{\frac{1}{2}} 0<\xi<1-\varepsilon, \\ &\quad (2-\xi-\eta)^{1/2}\leq\frac{1}{N(K_1, K_0, K_\alpha)}\Big(\theta- \frac{\nu(\nu+1)}{(\nu+\sigma)(\nu+\sigma+1)}\Big)\Big\}\,. \end{align*} It is possible that $D1=\emptyset$ or $D2=\emptyset$. Finally, for $\lambda >0$ we denote $$\label{Alambda} A_{\lambda }:=\frac{1}{\lambda (\lambda +1)}\max_{(\xi _0,\eta _0)\in \bar{D}_0^{(1)}}|(2-\xi _0-\eta _0)^{\lambda +2}F^{(i)}(\xi _0,\eta _0)|$$ and $$\label{C1F} C_{1}=\max \big\{A_{\nu +\sigma },\,\frac{\mu }{\nu +\sigma }A_{\mu } \max_{\overline{D1}}(2-\xi _0-\eta _0)^{-\mu +\nu +\sigma } \big\}\leq C_{\mu ,\sigma }\max_{\bar{D}_0^{(1)}}|F^{(i)}|,$$ where $C_{\mu ,\sigma }>0$ do not depend on $F^{(i)}$. If $D1=\emptyset$ we set $\max_{\overline{D1}} (\ldots )=1$. Now, we are ready to prove Theorem \ref{thmnew} by induction. \textbf{(i)} For $m=0$: \begin{gather*} U_{n,0}^{(i)}(\xi ,\eta )=p_{n,0}^{(i)}(\xi ,\eta )=q_{n,0}^{(i)}(\xi ,\eta )\equiv 0\text{ in }\bar{D}_{\varepsilon }^{(1)}, \\ E_{n,0}^{(i)}(\xi ,\eta )=F_{n}^{(i)}(\xi ,\eta ). \end{gather*} \textbf{(ii)} For $m=1$: \begin{align*} &(U_{n,1}^{(i)}-U_{n,0}^{(i)})(\xi _0,\eta _0) \\ &=\int_0^{\xi _0}\Big( \int_{\xi _0}^{\eta _0}E_{n,0}^{(i)}(\xi ,\eta )\,d\eta \Big) \,d\xi +2\int_0^{\xi _0}\Big( \int_0^{\eta }E_{n,0}^{(i)}(\xi ,\eta )\,d\xi \Big) \,d\eta \\ &=\int_0^{\xi _0}\Big( \int_{\xi _0}^{\eta _0}(2-\xi -\eta )^{-\lambda -2}(2-\xi -\eta )^{\lambda +2}F^{(i)}(\xi ,\eta )\,d\eta \Big) \,d\xi \\ &\quad +2\int_0^{\xi _0}\Big( \int_0^{\eta }(2-\xi -\eta )^{-\lambda -2}(2-\xi -\eta )^{\lambda +2}F^{(i)}(\xi ,\eta )\,d\xi \Big) \,d\eta . \end{align*} Applying Lemma \ref{lemma41} and recalling \eqref{Alambda}, we obtain $$|(U_{n,1}^{(i)}-U_{n,0}^{(i)})(\xi _0,\eta _0)|\leq A_{\lambda }(2-\xi _0-\eta _0)^{-\lambda }. \label{umu1}$$ Likewise we have $(p_{n,1}^{(i)}-p_{n,0}^{(i)})(\xi _0,\eta _0)=\int_0^{\xi _0}F^{(i)}(\xi ,\xi _0)d\xi +\int_{\xi _0}^{\eta _0}F^{(i)}(\xi _0,\eta )d\eta$ and with integration $$|(p_{n,1}^{(i)}-p_{n,0}^{(i)})(\xi _0,\eta _0)|\leq \lambda A_{\lambda }(2-\xi _0-\eta _0)^{-\lambda -1}, \label{pmu1}$$ respectively $$|(q_{n,1}^{(i)}-q_{n,0}^{(i)})(\xi _0,\eta _0)|\leq \lambda A_{\lambda }(2-\xi _0-\eta _0)^{-\lambda -1}. \label{qmu1}$$ For $\lambda =\nu +\sigma$ we have $$\begin{gathered} |(U_{n,1}^{(i)}-U_{n,0}^{(i)})(\xi_0,\eta_0)|\leq A_{\nu+\sigma}(2-\xi_0-\eta_0)^{-\nu-\sigma},\\ |(p_{n,1}^{(i)}-p_{n,0}^{(i)})(\xi_0,\eta_0)|\leq(\nu+\sigma) A_{\nu+\sigma}(2-\xi_0-\eta_0)^{-\nu-\sigma-1},\\ |(q_{n,1}^{(i)}-q_{n,0}^{(i)})(\xi_0,\eta_0)|\leq(\nu+\sigma) A_{\nu+\sigma}(2-\xi_0-\eta_0)^{-\nu-\sigma-1}. \end{gathered}\label{upqsigma1}$$ \textbf{(iii)} For $m=2,3,\ldots$ Now with Lemma \ref{lemma42}, the inequalities \eqref{umu1}--\eqref{qmu1} for $\lambda =\mu$ and induction, there exist sequences $\{U_{n,m}^{(i)}\}$, $\{p_{n,m}^{(i)}\}$ and $\{q_{n,m}^{(i)}\}$, $m\in \mathbb{N}$, of continuous functions and the estimates $$\begin{gathered} |(U_{n,m+1}^{(i)}-U_{n,m}^{(i)})(\xi_0,\eta_0)|\leq A_\mu(1-\delta_\nu)^m(2-\xi_0-\eta_0)^{-\mu},\\ |(p_{n,m+1}^{(i)}-p_{n,m}^{(i)})(\xi_0,\eta_0)|\leq\mu A_\mu(1-\delta_\nu)^m(2-\xi_0-\eta_0)^{-\mu-1},\\ |(q_{n,m+1}^{(i)}-q_{n,m}^{(i)})(\xi_0,\eta_0)|\leq\mu A_\mu(1-\delta_\nu)^m(2-\xi_0-\eta_0)^{-\mu-1} \end{gathered}\label{m+1estimates}$$ hold for $m=0,1,2,\ldots$. For the points $(\xi _0,\eta _0)\in D1$ from \eqref{m+1estimates} we obtain: \begin{align*} &|(U_{n,m+1}^{(i)}-U_{n,m}^{(i)})(\xi _0,\eta _0)| \\ &\leq A_{\mu }(1-\delta _{\nu })^{m}(2-\xi _0-\eta _0)^{-\nu -\sigma }\max_{\bar{D}1}{(2-\xi _0-\eta _0)^{-\mu +\nu +\sigma }}, \\ &|(p_{n,m+1}^{(i)}-p_{n,m}^{(i)})(\xi _0,\eta _0)| \\ &\leq (\nu +\sigma )\frac{\mu }{\nu +\sigma }A_{\mu }(1-\delta _{\nu })^{m}(2-\xi _0-\eta _0)^{-\nu -\sigma -1}\max_{\bar{D}1}{(2-\xi _0-\eta _0)^{-\mu +\nu +\sigma }}, \\ &|(q_{n,m+1}^{(i)}-q_{n,m}^{(i)})(\xi _0,\eta _0)| \\ &\leq (\nu +\sigma )\frac{\mu }{\nu +\sigma }A_{\mu }(1-\delta _{\nu })^{m}(2-\xi _0-\eta _0)^{-\nu -\sigma-1}\max_{\bar{D}1}{(2-\xi _0-\eta _0)^{-\mu +\nu +\sigma }}. \end{align*} Thus using \eqref{tau} and \eqref{C1F} in $D1$ for $m\in N$ we obtain $$\begin{gathered} |(U_{n,m+1}^{(i)}-U_{n,m}^{(i)})(\xi_0,\eta_0)|\leq C_1\tau^m(2-\xi_0-\eta_0)^{-\nu-\sigma},\\ |(p_{n,m+1}^{(i)}-p_{n,m}^{(i)})(\xi_0,\eta_0)|\leq(\nu+\sigma)C_1\tau^m(2- \xi_0-\eta_0)^{-\nu-\sigma-1},\\ |(q_{n,m+1}^{(i)}-q_{n,m}^{(i)})(\xi_0,\eta_0)|\leq(\nu+\sigma)C_1\tau^m(2- \xi_0-\eta_0)^{-\nu-\sigma-1}. \end{gathered} \label{estD1}$$ For $(\xi_0,\eta_0)\in D2$ we will show that such estimates hold too. Our induction hypothesis is that for some $m\in \mathbb{N}$ is true $$\begin{gathered}\label{estD2mm-1} |(U_{n,m}^{(i)}-U_{n,m-1}^{(i)})(\xi_0,\eta_0)|\leq C_1\tau^{m-1}(2-\xi_0-\eta_0)^{-\nu-\sigma},\\ |(p_{n,m}^{(i)}-p_{n,m-1}^{(i)})(\xi_0,\eta_0)|\leq(\nu+\sigma)C_1 \tau^{m-1}(2-\xi_0-\eta_0)^{-\nu-\sigma-1},\\ |(q_{n,m}^{(i)}-q_{n,m-1}^{(i)})(\xi_0,\eta_0)|\leq(\nu+\sigma)C_1 \tau^{m-1}(2-\xi_0-\eta_0)^{-\nu-\sigma-1} \end{gathered}$$ in $D_{\varepsilon }^{(1)}$, which for $m=1$ is fulfilled according to \eqref{upqsigma1} and $C_{1}\geq A_{\nu +\sigma }$. Now, we are trying to approve \eqref{estD1} in $D_{\varepsilon }^{(1)}$, which is already known in $D1$. By setting the inequalities \eqref{estD2mm-1} in \eqref{eqestEm} we derive \begin{align*} &|(E_{m}^{(i)}-E_{m-1}^{(i)})|(\xi ,\eta ) \\ &\leq \Big\{ \frac{\nu (\nu +1)}{(2-\xi -\eta )^{2}}+\frac{K_{1}}{2(2-\xi -\eta )}+\frac{K_0}{4}\Big\} C_{1}\tau ^{m-1}(2-\xi -\eta )^{-\nu -\sigma } \\ &\quad +\frac{(\nu +1/2)K_{1}}{2-\xi -\eta }C_{1}\tau ^{m-1}(2-\xi -\eta )^{-\nu -\sigma }\\ &\quad +\frac{3K_{1}}{2}(\nu +\sigma )C_{1}\tau ^{m-1}(2-\xi -\eta )^{-\nu -\sigma -1} \\ &\leq C_{1}\tau ^{m-1}\Big\{ \nu (\nu +1)(2-\xi -\eta )^{-\nu -\sigma -2}+ (5\nu +3\sigma +2)K_{1}(2-\xi -\eta )^{-\nu -\sigma -3/2} \\ &\quad +K_0(2-\xi -\eta )^{-\nu -\sigma -3/2}\Big\} \end{align*} everywhere in $D_{\varepsilon }^{(1)}$. Now we are ready to apply Lemma \ref{lemma41} for all the terms in the brackets, since they are of power less than $-2$. We substitute the last inequality in the formulas \eqref{eqappr} and with integration and Lemma \ref{lemma41} we obtain: \begin{align*} &|(U_{n,m+1}^{(i)}-U_{n,m}^{(i)})(\xi _0,\eta _0)|\\ &\leq C_{1}\tau ^{m-1}\Big\{ \frac{\nu (\nu +1)}{(\nu +\sigma )(\nu +\sigma +1)} (2-\xi _0-\eta _0)^{-\nu -\sigma } \\ &\quad +\frac{(5\nu +3\sigma +2)K_{1}+K_0}{(\nu +\sigma-1/2 )(\nu +\sigma+1/2)}(2-\xi _0-\eta _0)^{-\nu -\sigma +1/2} \Big\} \\ &\quad +K_{\alpha }\int_0^{\xi _0}|(U_{n,m}^{(i)}-U_{n,m-1}^{(i)})(\xi ,\xi )|d\xi \\ &\leq C_{1}\tau ^{m-1}(2-\xi _0-\eta _0)^{-\nu -\sigma }\Big\{ \frac{\nu (\nu +1)}{(\nu +\sigma )(\nu +\sigma +1)}\\ &\quad +(2-\xi _0-\eta _0)^{1/2}N(K_{1},K_0,K_{\alpha })\Big\} , \end{align*} \begin{align*} &|(p_{n,m+1}^{(i)}-p_{n,m}^{(i)})(\xi _0,\eta _0)|\\ &\leq (\nu +\sigma)C_{1}\tau ^{m-1} \Big\{ \frac{\nu (\nu +1)}{(\nu +\sigma )(\nu +\sigma +1)} (2-\xi _0-\eta _0)^{-\nu -\sigma -1} \\ &\quad +\frac{(5\nu +3\sigma +2)K_{1}+K_0}{(\nu +\sigma-1/2 )(\nu +\sigma+1/2) }(2-\xi _0-\eta _0)^{-\nu -\sigma -1/2} \Big\} \\ &\quad +2K_{\alpha }C_{1}\tau ^{m-1}(2-2\xi _0)^{-\nu -\sigma-1/2 } \\ &\leq (\nu +\sigma )C_{1}\tau ^{m-1}(2-\xi _0-\eta _0)^{-\nu -\sigma -1} \Big\{ \frac{\nu (\nu +1)}{(\nu +\sigma )(\nu +\sigma +1)}\\ &\quad +(2-\xi_0-\eta _0)^{1/2}N(K_{1},K_0,K_{\alpha })\Big\} , \end{align*} \begin{align*} &|(q_{n,m+1}^{(i)}-q_{n,m}^{(i)})(\xi _0,\eta _0)|\\ &\leq (\nu +\sigma)C_{1}\tau ^{m-1}(2-\xi _0-\eta _0)^{-\nu -\sigma -1} \Big\{ \frac{\nu (\nu +1)}{(\nu +\sigma )(\nu +\sigma +1)}\\ & \quad +(2-\xi_0-\eta _0)^{1/2} N(K_{1},K_0,K_{\alpha })\Big\} \end{align*} in $D_{\varepsilon }^{(1)}$. Since $\frac{\nu (\nu +1)}{(\nu +\sigma )(\nu +\sigma +1)}+(2-\xi _0-\eta _0)^{1/2}N(K_{1},K_0,K_{\alpha })\leq \theta \leq \tau \quad\text{in } D2$ by definition, for the points $(\xi_0,\eta _0)\in D2$ from the last three inequalities we obtain \eqref{estD1}. Then by induction we conclude that the estimates \eqref{estD1} hold in $D_{\varepsilon }^{(1)}$ for $m=2,3,\ldots$. The functions $\{U_{n,m}^{(i)},p_{n,m}^{(i)},q_{n,m}^{(i)}\}_{m=0}^{\infty }$ belong to $C(\bar{D}_{\varepsilon }^{(1)})$ and we have uniform convergence to some functions $\{U_{n}^{(i)},p_{n}^{(i)},q_{n}^{(i)}\}\in C(\bar{D} _{\varepsilon }^{(1)})$, as $m\to \infty$ and \begin{align*} |U_{n}^{(i)}(\xi _0,\eta _0)| &=\big|\sum_{m=0}^{\infty}(U_{n,m+1}^{(i)}-U_{n,m}^{(i)})(\xi _0,\eta _0)\big| \\ &\leq C_{1}(1-\tau )^{-1}(2-\xi _0-\eta _0)^{-\nu -\sigma }, \\ |U_{n,\xi _0}^{(i)}(\xi _0,\eta _0)| &=\big|\sum_{m=0}^{\infty }(p_{n,m+1}^{(i)}-p_{n,m}^{(i)})(\xi _0,\eta _0)\big| \\ &\leq (\nu +\sigma )C_{1}(1-\tau )^{-1}(2-\xi _0-\eta _0)^{-\nu -\sigma -1}, \\ |U_{n,\eta _0}^{(i)}(\xi _0,\eta _0)| &=\big|\sum_{m=0}^{\infty}(q_{n,m+1}^{(i)}-q_{n,m}^{(i)})(\xi _0,\eta _0)\big| \\ &\leq (\nu +\sigma )C_{1}(1-\tau )^{-1}(2-\xi _0-\eta _0)^{-\nu -\sigma -1}. \end{align*} In view of \eqref{C1F}, these estimates coincide with \eqref{newestimates} with $C_{\sigma }=C_{\mu,\sigma}(1-\tau )^{-1}$. \end{proof} \begin{proof}[Proof of Theorem \ref{mainres}] First, we note that the conditions (i) and (ii) of Theorem \ref{thm44} are fulfilled, hence we can apply Theorem \ref{thmnew}. Using the relations \eqref{eqU} and \eqref{charcoord}, we make the inverse transformation from Problem $P_{\alpha,2}$ to Problem $P_{\alpha }$ and we see that the generalized solution $u(x,t)$ belongs to $C^{1}(\bar{\Omega}_0\backslash O)$ and the estimates \begin{gather*} |u(x,t)|\leq C_{n,\sigma }\max_{\bar{\Omega}_0}{ \{|f_{n}^{(1)}|+|f_{n}^{(2)}|\}}|x|^{-n-\sigma }, \\ \sum_{|\beta |=1}|D^{\beta }u(x,t)|\leq n C_{n,\sigma }\max_{\bar{\Omega} _0}{\{|f_{n}^{(1)}|+|f_{n}^{(2)}|\}} |x|^{-n-\sigma -1} \end{gather*} hold, where $C_{n,\sigma }>0$ depends on $n$, $\sigma$ and all coefficients of \eqref{eq0p1}. \end{proof} It is easy to generalize this result in the following way. \begin{theorem}\label{thcomn} Let the right-hand function $f(\varrho,\varphi,t)$ of \eqref{eq0polar} be a trigonometric polynomial $$\label{trigp} f=\sum_{n=0}^l f_n^{(1)}(\varrho,t)\cos n\varphi+f_n^{(2)}(\varrho,t)\sin n\varphi, \quad l\in\mathbb{N}.$$ If conditions (i) and (ii) of Theorem \ref{thm44} are fulfilled, then there exists one and only one generalized solution $u(x,t)\in C^1(\bar{\Omega}_0\backslash O)$ of Problem $P_{\alpha }$ and the a priori estimates \begin{gather*} |u(x,t)|\leq C_{l,\sigma}\max_{\bar{\Omega}_0}{\{|f_{l}^{(1)}|+|f_{l}^{(2)}|\}} |x|^{-l-\sigma}+O(|x|^{-l-\sigma+1}), \\ \sum_{|\beta|=1}|D^\beta u(x,t)|\leq C_{l,\sigma} \max_{\bar{\Omega}_0}{\{|f_{l}^{(1)}|+|f_{l}^{(2)}|\}} |x|^{-l-\sigma-1}+O(|x|^{-l-\sigma}) \end{gather*} hold. \end{theorem} \section{On the singularity of solutions of problem $\boldsymbol{P_{\alpha,2}}$} In this section we derive some sufficient conditions on the coefficients and the right-hand side of \eqref{eq0p1} for the existence of singular solutions of the problem we treat. We follow Grammatikopoulos et al \cite{GHP} (see Theorem \ref{thmG}) and making some modifications we extend this result. First, we represent an important lemma \begin{lemma} [\cite{GHP}] \label{lemmamax} Consider Problem $P_{\alpha,2}$. Let $F^i, A_i, B_i, C_i, D_i\in C({\bar{D}}_{\varepsilon }^{(1)})$, $i=1,2$, $$A_i\geq 0,\quad B_i\geq 0,\quad C_i\geq 0,\quad D_i\geq 0,\quad \alpha (1-\xi )\geq 0 \quad \text{ in }\bar{D}_{\varepsilon }^{(1)}, \quad i=1,2 \label{ABCDgeq0}$$ and $$F^{(i)}\geq 0\quad \text{in } \bar{D}_{\varepsilon }^{(1)},\quad i=1,2. \label{Feq0}$$ Then for the solution $(U^{(1)},U^{(2)})$ of Problem $P_{\alpha ,2}$ we have $$\label{Ugeq0} U^{(i)}(\xi ,\eta )\geq 0,\quad U_{\eta }^{(i)}(\xi ,\eta )\geq 0,\quad U_{\xi }^{(i)}(\xi ,\eta )\geq 0\quad \text{for }(\xi ,\eta )\in \bar{D}_{\varepsilon }^{(1)},\; i=1,2.$$ \end{lemma} Note that in view of $D_1=-D_2$ (see \eqref{eqABCD}) for \eqref{ABCDgeq0} to be fulfilled is necessary $D_1=D_2\equiv0$, so in this case we may consider the system \eqref{eqdarbu} as two independent single equations $$\label{darbusingle} U_{\xi \eta }-AU_{\xi }-BU_{\eta }-CU=F(\xi ,\eta )$$ with boundary conditions $$\label{bound2} U(0,\eta )=0,\quad (U_{\eta }-U_{\xi })(\xi ,\xi )+\alpha (1-\xi )U(\xi ,\xi )=0.$$ Next, we formulate the main result in this section. \begin{theorem} \label{thmsingul} Consider the problem \eqref{darbusingle}, \eqref{bound2}. Let for the coefficients we assume $A,B,C\in C(\bar{D}_{\varepsilon }^{(1)}), \alpha(1-\xi) \in C^1([0,1-\varepsilon])$ and $$\label{somcon} A\geq 0,\quad B\geq 0,\quad C\geq\frac{4n^2-1}{4(2-\xi -\eta )^2},\quad \alpha (1-\xi )\geq 0 \text{ in {\bar{D}}_{\varepsilon }^{(1)}}.$$ Additionally, let $F(\xi,\eta)\in C(\bar{D}_{\varepsilon }^{(1)})$ does not change its sign (that means either $F\geq0$ or $F\leq0$) and $F\not\equiv0$ in $D_0^{(1)}$. Then for $\eta\in(0,1]$ and $\varepsilon\in(0,\varepsilon_F)$, where $\varepsilon_F\in(0,1)$ is a number depending on $F$, holds $$\label{geqest} |U(1-\varepsilon,\eta)|\geq C_0\varepsilon^{-(n-\frac{1}{2})},\quad C_0=\text{const}>0.$$ \end{theorem} \begin{proof} We will consider the case $F\geq0$. The case $F\leq0$ is obviously analogous. In \cite{GHP} was shown the existence of classical solution $U(\xi,\eta)$ of the problem we treat. We introduce a function $W(\xi,\eta):=\frac{(1-\xi)^{n-1/2}(1-\eta)^{n-1/2}}{(2-\xi-\eta)^{n-1/2}}.$ We see that $W(\xi,\eta)>0$ in $D_\varepsilon^{(1)}$. Next, since $F\not\equiv0$ in $D_0^{(1)}$ and it is continuous in each $D_\varepsilon^{(1)}$, we conclude that there exists an open ball in $D_0^{(1)}$ where $F>0$. Therefore, if we consider $\varepsilon$ small enough (smaller than some $\varepsilon_F$), we have the inequality $$\label{K} \int_{D_\varepsilon}(FW)(\xi,\eta)d\xi d\eta\geq K,\quad K=\text{const}>0.$$ Recall that $D_{\varepsilon }=\{(\xi ,\eta ):0<\xi <\eta <1-\varepsilon \}$, $D_{\varepsilon }\subset D_{\varepsilon }^{(1)}$. Using \eqref{darbusingle} we transform \eqref{K} in the following way: \begin{align*} 0