Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 178, pp. 1-20.
Title: Existence and blow-up of solutions for a semilinear filtration problem
Authors: Evangelos A. Latos (Univ. of Graz, Austria)
Dimitrios E. Tzanetis (National Technical Univ. of Athens, Greece)
Abstract:
We first examine the existence and uniqueness of local solutions
to the semilinear filtration equation $u_t=\Delta K(u)+\lambda f(u)$,
for $\lambda>0$, with initial data $u_0\geq 0$ and appropriate boundary
conditions. Our main result is the proof of blow-up of solutions
for some $\lambda$.
Moreover, we discuss the existence of solutions for the corresponding
steady-state problem. It is found that there exists a critical value
$\lambda^*$ such that for $\lambda>\lambda^*$ the problem has no
stationary solution of any kind, while for $\lambda\leq\lambda^*$
there exist classical stationary solutions.
Finally, our main result is that the solution for $\lambda>\lambda^*$,
blows-up in finite time independently of $u_0\geq0$. The functions
$f,K$ are positive, increasing and convex and $K'/f$ is integrable
at infinity.
Submitted July 14, 2013. Published August 04, 2013.
Math Subject Classifications: 35K55, 35B44, 35B51, 76S05.
Key Words: Blow-up; filtration problem; existence; upper and lower solutions.