The diffusion phenomenon for damped wave equations with space-time dependent coefficients
The paper investigates the long-time behavior of solutions for IVP to damped wave equations of the form \(u_{tt}+u_t-\nabla\cdot (a(x,t)\nabla u)=0\) with initial data \(u_0(x)\), \(u_1(x)\), \(x\in \mathbb{R}^N\). Decay estimates in \({L_x}^2\) norm for the solution, its gradient and its derivative with respect to \(t\) are obtained and a large discussion on improved decay is done. Results on exponentially decay outside a ball of the derivatives of the solution are shown. Under some special assumptions one proves that the solution \(u(x, t)\) behaves like the solution to the diffusion problem \(v_t-\nabla\cdot (a(x,t)\nabla v)=0\), \(v(x,0)=u_0(x)+u_1(x)\). In the acknowledgments part, the author writes: ``I would like to ask Kat Ashdown, the most caring person I have ever known, to marry me''.
scientific article; zbMATH DE number 6951278
scientific article