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.