On the long time behaviour of solutions to dissipative wave equations in $$\mathbb{R}^{2} $$

Abstract

In this paper we present a parabolic approach to studying the diffusive long time behaviour of solutions to the Cauchy problem:where u0 and u1 satisfy suitable assumptions. After an appropriate scaling we obtain the convergence to a stationary solution in L q norm (1 ≤ q < ∞).2000 Mathematics Subject Classification: 35L15; 35B40.

Asymptotic expansion of L2-norms of solutions to the heat and dissipative wave equations

“…The diffusion phenomenon for linear and nonlinear wave equations has been investigated in many papers. Nishihara [10] (see also [1,3,7,9] derives L p -L q estimates of the form…”