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

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 $u_0geq 0$ 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 $lambdaleqlambda^*$ there exist classical stationary solutions. Finally, our main result is that the solution for $lambda>lambda^*$, blows-up in finite time independently of $u_0geq0$. The functions $f,K$ are positive, increasing and convex and $K'/f$ is integrable at infinity.

Keywords

• Blow-up
• filtration problem
• existence
• upper and lower solutions