Continuation beyond interior gradient blow-up in a semilinear parabolic equation

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 . . .

Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory

with a parameter β 0, which is a model equation from explosion-convection theory. Unlike the classical Frank-Kamenetskii equation ut = uxx+e u (a solid fuel model), by using analytical and numerical evidence, we show that the generic blow-up in this fourth-order problem is described by a similarity solution u∗(x, t) = − ln(T − t) + f1(x/(T − t)) (T > 0 is the blowup time), with a non-trivial pr...

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...

Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations

We study the Cauchy problem in R × R+ for one-dimensional 2mth-order, m > 1, semilinear parabolic PDEs of the form (Dx = ∂/∂x) ut = (−1) D x u + |u| u, where p > 1, and ut = (−1) D x u + e u with bounded initial data u0(x). Specifically, we are interested in those solutions that blow up at the origin in a finite time T . We show that, in contrast to the solutions of the classical secondorder pa...