Regularity in Sobolev spaces for doubly nonlinear parabolic equations
Let \(m>1\) and \(p>(m+1)/m\) be constants, and define \(A(\xi)=|\xi|^{p-2}\xi\) and \(a(z)=|z|^{m-1}z\). The authors examine the regularity of solutions of the Cauchy problem \[ u_{t} =\operatorname {div} A(D[a(u)])\quad \text{in } \mathbb R^{n}\times(0,T), \qquad u(x,0)=u_0(x) \quad\text{in } \mathbb R^{n} \] for \(n \geq 2\) and \(T>0\). If \(q<\max\{p/(p+1),2/3\}\) and \(r<1/q\), it is shown that \[ a(u) \in L^{q}(0,T;W^{r,q}(\mathbb R^{n})). \] Similar estimates are proved for powers of \(|Da(u)|\). The proof is based on difference quotient arguments developed by the first author [Z. Anal. Angew. 15, 637-650 (1996; Zbl 0858.35069)] for elliptic \(p\)-Laplace type equations.
scientific article; zbMATH DE number 1884227
scientific article