Sobolev regularity for \(t>0\) in quasilinear parabolic equations (Q2770429)

The author establishes a regularity property for the solutions to the quasilinear parabolic initial-boundary value problem NEWLINE\[NEWLINE \left\{\begin{aligned} v_t-a_{ij}(\nabla v) \partial_i\partial_j v=g(x,t) & \quad \text{ in}\quad \Omega\times(0,T),\\v(x,0)=v_0(x) & \quad \text{ in}\quad \Omega\times \{t=0\},\\ v(\cdot, t)=0 & \quad \text{ in}\quad \partial\Omega\times (0,T).\end{aligned}\right. NEWLINE\]NEWLINE It is shown that for \(t>0\) the solutions belong to the same space to which the solutions of the following second-order hyperbolic problem NEWLINE\[NEWLINE \left\{\begin{aligned} \varepsilon u_{tt}+u_t-a_{ij}(\nabla u) \partial_i\partial_j u=f(x,t) & \quad \text{ in}\quad \Omega\times(0,T),\\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x) & \quad \text{ in} \quad \Omega\times \{t=0\},\\ u(\cdot, t)=0 & \quad \text{ in} \quad \partial\Omega\times (0,T)\end{aligned}\right. NEWLINE\]NEWLINE which is a singular perturbation of the parabolic one. The result provides another illustration of the asymptotically parabolic nature of the hyperbolic problem and is needed to establish the diffusion phenomenon for quasilinear dissipative wave equations in Sobolev spaces.