Electron. J. Diff. Equ., Vol. 2013 (2013), No. 178, pp. 1-20.

Existence and blow-up of solutions for a semilinear filtration problem

Evangelos A. Latos, Dimitrios E. Tzanetis

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 4, 2013.
Math Subject Classifications: 35K55, 35B44, 35B51, 76S05.
Key Words: Blow-up; filtration problem; existence; upper and lower solutions.

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Evangelos A. Latos
Institute for Mathematics and Scientific Computing
University of Graz
A-8010 Graz, Heinrichstrasse, 36, Austria
email: evangelos.latos@uni-graz.at
Dimitrios E. Tzanetis
Department of Mathematics
School of Applied Mathematical and Physical Sciences
National Technical University of Athens, Zografou Campus
157 80 Athens, Greece
email: dtzan@math.ntua.gr

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