The blow-up rate for a non-scaling invariant semilinear wave equations in higher dimensions

TL;DR

This paper extends the understanding of blow-up rates for a class of semilinear wave equations with logarithmic modifications in higher dimensions, showing they match the rates predicted by associated ordinary differential equations.

Contribution

It generalizes previous one-dimensional results to higher dimensions for semilinear wave equations with a specific nonlinearity involving a logarithmic factor.

Findings

Blow-up rate matches the ODE solution rate in higher dimensions.

Extension of one-dimensional blow-up results to higher dimensions.

Provides a detailed analysis of singular solutions for the given wave equation.

Abstract

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We will show that the blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, The result in one space dimension, has been proved in \cite{HZjmaa2020}. Our goal here is to extend this result to higher dimensions.