Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Abstract

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type $$ u_t= -Δ^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, $$ are discussed. For the semilinear heat equation $u_t=\Delta u+u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.

41 pages, 27 figures

References in corpus (5)

• Bright cavity polariton solitons
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• Supercritical biharmonic equations with power-type nonlinearity
• Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective
• On blow-up "twistors" for the Navier--Stokes equations in $R^3$: a view from reaction-diffusion theory

Cited by in corpus (5)

• Multiple Quenching Solutions of a Fourth Order Parabolic PDE with a singular nonlinearity modelling a MEMS Capacitor
• On blow-up "twistors" for the Navier--Stokes equations in $R^3$: a view from reaction-diffusion theory
• Moving Mesh simulation of contact sets in two dimensional models of elastic-electrostatic deflection problems
• Boundary Characteristic Point Regularity for Navier-Stokes Equations: Blow-up Scaling and Petrovskii-type Criterion (a Formal Approach)
• Incomplete self-similar blow-up in a semilinear fourth-order reaction-diffusion equation