Incomplete Self-Similar Blow-Up in a Semilinear Fourth-Order Reaction-Diffusion Equation

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

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...

Five Types of Blow-up in a Semilinear Fourth-order Reaction-diffusion Equation: an Analytic-numerical Approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type ut = −∆2u+ |u|p−1u in R × (0, T ), p > 1, limt→T− supx∈RN |u(x, t)| = +∞, are discussed. For the semilinear heat equation ut = ∆u+ u , various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide r...

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...

Self-similar blow-up for a diffusion–attraction problem

In this paper we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see, e.g., Brenner M P et al 1999 Nonline...