Uniform blow-up estimates for nonlinear heat equations and applications

Blow-up in Nonlinear Heat Equations

In this paper we study the blowup problem of nonlinear heat equations. Our result show that for a certain family of initial conditions the solution will blowup in finite time, the blowup parameters satisfy some dynamics which are asymptotic stable, moreover we provide the remainder estimates. Compare to the previous works our approach is analogous to one used in bifurcation theory and our techn...

Blow-up at space infinity for nonlinear heat equations

Blow-Up Time for Nonlinear Heat Equations with Transcendental Nonlinearity

where x ∈ R, F is a given nonlinear function and u is unknown. Due to the mathematical and physical importance, existence and uniqueness theories of solutions of nonlinear heat equations have been extensively studied by many mathematicians and physicists, for example, 1–10 and references therein. Unlike other studies, we focus on the nonlinear heat equations with transcendental nonlinearities s...

The Blow-up Rate Estimates for a System of Heat Equations with Nonlinear Boundary Conditions

This paper deals with the blow-up properties of positive solutions to a system of two heat equations ut = ∆u, vt = ∆v in BR× (0, T ) with Neumann boundary conditions ∂u ∂η = e vp , ∂v ∂η = e uq on ∂BR × (0, T ), where p, q > 1, BR is a ball in Rn, η is the outward normal. The upper bounds of blow-up rate estimates were obtained. It is also proved that the blow-up occurs only on the boundary.

On uniform nullcontrollability and blow-up estimates

The paper looks at some of the techniques (separation of variables, Fourier series, Carleman estimates) used to obtain observability and nullcontrollability results to determine qualitatively the asymptotic behavior of the estimates obtained with respect to relevant parameters. We are particularly concerned with blow-up as the control time becomes short (T → 0).