$L_p$-$L_q$-theory for a quasilinear non-isothermal Westervelt equation

TL;DR

This paper develops an $L_p$-$L_q$-theory for a combined quasilinear system involving the Westervelt equation and Pennes bioheat equation, proving well-posedness, regularization, and analyzing long-term dynamics.

Contribution

It introduces a novel $L_p$-$L_q$-framework for this coupled system, establishing well-posedness and solution regularization results.

Findings

Proved local and global well-posedness of the system.

Showed solutions regularize instantaneously.

Analyzed equilibria and long-term behavior.

Abstract

We investigate a quasilinear system consisting of the Westervelt equation from nonlinear acoustics and Pennes bioheat equation, subject to Dirichlet or Neumann boundary conditions. The concept of maximal regularity of type $L_p$-$L_q$ is applied to prove local and global well-posedness. Moreover, we show by a parameter trick that the solutions regularize instantaneously. Finally, we compute the equilibria of the system and investigate the long-time behaviour of solutions starting close to equilibria.