Well-posedness results and blow-up for a class of semilinear heat equations

Dang Van Yen,Ho Thi Kim Van,Le Dinh Long,Ho Duy Binh

doi:10.1186/s13662-021-03404-x

Dang Van Yen, Ho Thi Kim Van + Show 2 more

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https://doi.org/10.1186/s13662-021-03404-x

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Abstract

This paper considers the initial value problem for nonlinear heat equation in the whole space mathbb{R}^{N}. The local existence theory related to the finite time blow-up is also obtained for the problem with nonlinearity source (like polynomial types).

Highlights

We consider the following initial value problem for u : RN × [0, T] → R: ⎧⎨ut = u + λu – ρum, in RN × [0, T],⎩u(x, 0) = f (x), in RN, (P)where λ, ρ ∈ R and m, N ∈ N∗ are parameters; is the standard Laplacian with Dirichlet boundary conditions in L2(RN ); u = u(x, t) is the state of the unknown function and f is given function
The goal of this paper is the study of the local existence, unique continuation and a finite time blowup of solution to Problem (P)
Given > 0 there exists N ∈ N such that u(·, tn) – u(·, tk) H2s(RN )

Summary

Introduction

The goal of this paper is the study of the local existence, unique continuation and a finite time blowup of solution to Problem (P). We establish the results on the continuation and finite time blow-up of solutions in H2s(RN ). 3; we present local existence, uniqueness continuation of the solution is discussed in Sect.

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