\(L^1\) decay estimates for dissipative wave equations (Q2725353)

If one is interested in the asymptotic behaviour for \(t\to \infty\) of solutions to the Cauchy problem for the dissipative wave equation NEWLINE\[NEWLINEu_{tt}+u_t-\Delta u=0,\quad u(\cdot,0)=u_0,\quad u_t(\cdot,0)=u_1NEWLINE\]NEWLINE and to the Cauchy problem for the heat equation NEWLINE\[NEWLINEv_t - \Delta v = 0,\quad v(\cdot,0) =v_0,NEWLINE\]NEWLINE then these behaviours coincide. This coincidence was studied by Matsumura and decay estimates for the \(L_p\)-norms of \(\partial^k_t \partial^\alpha_x u\) and \(\partial^k_t\partial^\alpha_x v\) for \(p\in [2, +\infty]\) were given under the assumption that the data \(u_0,u_1\) belong to Sobolev spaces \(H^r\) and to \(L_1\), where \(r\) depends in a natural way on \(k\), \(\alpha\) and \(n\). Moreover, in a previous paper the authors showed the validity of a diffusion phenomenon, that is, the \(L_\infty\)-norm of \(u-v\) has a better decay order as those of \(u\) and \(v\) separately.NEWLINENEWLINENEWLINEThe present paper is devoted to attack decay estimates for the \(L_1\)-norm of \(\partial^k_t\partial^\alpha_x u\), \(\partial^k_t\partial^\alpha_x v\) and to discuss their coincidence. To reach this goal the authors assume additionally that \(u_0,u_1\) belong to weighted \(L_1\)-spaces \(L^s_1\), where \(s\) depends on \(k\), \(\alpha\) and \(n\), too. It would be interesting to understand if the order \(s\) is sharp. Thus the authors could close the gap \(p\in [1,2)\) under special assumptions.NEWLINENEWLINENEWLINEThe main tools are the explicit representation of the solution to the Cauchy problem by Fourier multipliers, theory of special functions and a careful division of the integration set to estimate \(\|\partial^k_t\partial^\alpha_x u\|_{L_1}\).