$$L_p$$–$$L_q$$-Theory for a Quasilinear Non-isothermal Westervelt Equation

Abstract

AbstractWe 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 p –$$L_q$$ 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.