Van Tien Nguyen

Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations

We construct a solution for a class of strongly perturbed semilinear heat equations which blows up in finite time with a prescribed blow-up profile. The construction relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude.

Blow-up results for a strongly perturbed semilinear heat equation: theoretical analysis and numerical method

We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this Lyapunov functional, we derive the blow-up rate and the blow-up limit of the solution. We also classify all asymptotic behaviors of the solution at the singularity and give precisely blow-up profiles corresponding to these behaviors. Finally, we attain the blow-up profile numerically, thanks to a new mesh-refinement algorithm inspired by the rescaling method of Berger and Kohn in 1988. Note that our method is applicable to more general equations, in particular those with no scaling invariance.

On the blow-up results for a class of strongly perturbed semilinear heat equations

We consider in this work some class of strongly perturbed for the semilinear heat equation with Sobolev sub-critical power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. We also classify all possible asymptotic behaviors of the solution when it approaches to singularity. Finally, we describe precisely the blow-up profiles corresponding to these behaviors.

Numerical analysis of the rescaling method for parabolic problems with blow-up in finite time

Abstract In this work, we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation u ↦ u λ ( x , t ) : = λ 2 p − 1 u ( λ x , λ 2 t ) . For that purpose, we apply the rescaling method proposed by Berger and Kohn (1988) to such problems. The convergence of the method is proved under some regularity assumption. Some numerical experiments are given to derive the blow-up profile verifying henceforth the theoretical results.

Refined Regularity of the Blow-Up Set Linked to Refined Asymptotic Behavior for the Semilinear Heat Equation

Abstract We consider u ⁢ ( x , t )

Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

{u(x,t)}

Construction and stability of blowup solutions for a non-variational semilinear parabolic system

, a solution of ∂ t ⁡ u = Δ ⁢ u + | u | p - 1 ⁢ u

Finite degrees of freedom for the refined blow-up profile of the semilinear heat equation

{\partial_{t}u=\Delta u+|u|^{p-1}u}

Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation

which blows up at some time T > 0

On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

{T>0}