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:

$$ \left\{ \begin{aligned} & u_{{ t t }} + u_{t} - \Delta u = 0,x \in R^{N} ,t \ > 0,\\ & u(\cdot, 0) = u_{0} ,x \in R^{N} ,\\ & u(\cdot, 0) = u_{1} ,x \in R^{N} x; \end{aligned} \right. $$

(1)

where u₀ and u₁ satisfy suitable assumptions.

After an appropriate scaling we obtain the convergence to a stationary solutio n in L^q norm (1 ≤  q  <  ∞).