Fujita modified exponent for scale invariant damped semilinear wave equations

Felisia Angela Chiarello,Giovanni Girardi,Sandra Lucente

doi:10.1007/s00028-021-00705-2

Abstract

The aim of this paper is to prove a blow-up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition, we give an upper bound estimate for the lifespan of the solution. It depends not only on the exponent of the nonlinear term and not only on the damping coefficient but also on the size of the decay rate of the initial data.

Highlights

In the recent years, the following Cauchy problem for the wave equation with scale invariant damping spreads a new line of research on variable coefficient-type equations
The paper is organized as follows: In Sect. 2, we give an overview of the known results and we state an auxiliary theorem; in Sect. 3, we prove the main results
In the even case [10], the initial data satisfy (5) for k ∈ (k1(n, p, μ), k2(n, p, μ)] such that k1(n, p, μ) = max n k2(n, p, μ) = min n − 1, p−. Rewriting these conditions in terms of p, we find that p > pF

Summary

Introduction

The following Cauchy problem for the wave equation with scale invariant damping spreads a new line of research on variable coefficient-type equations. In [2,3,10,11], some results on the global existence of a solution for (1) with non-compactly initial data appeared assuming a suitable decay behavior for g. The main point is to find a critical exponent, fixed a suitable space of data. Changing the space of data, a change of critical exponent may appear. The novelty of our result consists in showing that if one takes into account the decay rate of the initial data, the Fujita-type exponent depends on such decay rate. We give an upper bound estimate for the lifespan of the solution, in terms of the power of the nonlinear term, the size and the growth of the initial data. The paper is organized as follows: In Sect. 2, we give an overview of the known results and we state an auxiliary theorem; in Sect. 3, we prove the main results

Motivations

Proof of Theorem 2

Proof of Theorem 1