\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 50, pp. 1--7.\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-2012/50\hfil Dynamic equations on time scales] {Integral inequalitys for partial dynamic equations on time scales} \author[D. B. Pachpatte\hfil EJDE-2012/50\hfilneg] {Deepak B. Pachpatte} \address{Deepak B. Pachpatte \newline Department of Mathematics, Dr. Babasaheb Ambedekar Marathwada University, Aurangabad, Maharashtra 431004, India} \email{pachpatte@gmail.com} \thanks{Submitted January 16, 2012. Published March 27, 2012.} \subjclass[2000]{26E70, 34N05} \keywords{Dynamic equations; time scales; qualitative properties;\hfill\break\indent inequalities with explicit estimates} \begin{abstract} The aim of the present paper is to study some basic qualitative properties of solutions of some partial dynamic equations on time scales. A variant of certain fundamental integral inequality with explicit estimates is used to establish our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} %\newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} During past few years many authors have established the time scale analogue of well known dynamic equations used in the development of theory of differential and integral equation see \cite{Boh3,D1,D2,D3,D4,W2,S1,Z2}. In \cite{Boh4,Boh5,H1,J1,A1} authors have obtained some results on multiple integration and partial dynamic equations on time scales. Recently in \cite{O1,A2,A3,W1} authors have obtained inequalities on two independent variables on time scales. In the present paper we establish some basic qualitative properties of solutions of some partial dynamic equation on time scales. We use certain fundamental integral inequality with explicit estimates to establish our results. We assume understanding of time scales and its notation. Excellent information about introduction to time scales can be found in \cite{Boh1,Boh2}. In what follows $\mathbb{R}$ denotes the set of real numbers, $\mathbb{Z}$ the set of integers and $\mathbb{T}$ denotes arbitrary time scales. Let $C_{rd}$ be the set of all rd continuous function. We assume $\mathbb{T}_1$ and $\mathbb{T}_2$ be two time scales and $\Omega=\mathbb{T}_1 \times \mathbb{T}_2$. In this article, we consider partial dynamic equation of the type $$u^{\Delta{t}} (t,x) = f({t,x,u(t,x)}) + \int_{s_0}^s {g({t,x,y,u({t,y})})\Delta y + h(t,x)} ,\label{e1.1}$$ which satisfies the initial condition $$u(t_0,x)=u_0(x), \label{e1.2}$$ for $({t_0 ,x}) \in \Omega$, where $u_0 \in C(I,\mathbb{R})$, $I=[a,b]$ \$(a