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| We first examine the local existence and uniqueness of solutions $u=u(x,t;\lambda)$ to the semi linear filtration equation $u_t=\Delta K(u)+\lambda f(u)$, for $\lambda>0$, with initial data $u_0\geq0$ 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 $u$, for $\lambda>\lambda^*$, blows-up in finite time independently of $u_0\geq0$. The functions $f,K$ are mostly positive, increasing and convex and $K'/f$ is integrable at infinity. |
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