Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium

Initial Blow-up of Solutions of Semilinear Parabolic Inequalities

We study classical nonnegative solutions u(x, t) of the semilinear parabolic inequalities 0 ≤ ut −∆u ≤ u in Ω× (0, 1) where p is a positive constant and Ω is a bounded domain in R, n ≥ 1. We show that a necessary and sufficient condition on p for such solutions u to satisfy a pointwise a priori bound on compact subsets K of Ω as t→ 0 is p ≤ 1 + 2/n and in this case the bound on u is max x∈K u(x...

Blow-up estimates for a semilinear coupled parabolic system

This work deals with a semilinear parabolic systemwhich is coupled both in the equations and in the boundary conditions. The blow-up phenomena of its positive solutions are studied using the scaling method, the Green function and Schauder estimates. The upper and lower bounds of blow-up rates are then obtained. Moreover we show the influences of the reaction terms and the boundary absorption te...

Interior Gradient Blow-up in a Semilinear Parabolic Equation

We present a one dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single...

Blow-Up in a Fourth-Order Semilinear Parabolic Equation From . . .

The Blow-up Problem for a Semilinear Parabolic Equation with a Potential

Let Ω be a bounded smooth domain in R . We consider the problem ut = ∆u + V (x)u in Ω × [0, T ), with Dirichlet boundary conditions u = 0 on ∂Ω × [0, T ) and initial datum u(x, 0) = Mu0(x) where M ≥ 0, u0 is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of M . As a consequence of these estimates we find that, for M larg...