Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity $q\ge 2$

Fattorusso, Luisa

Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 73-90.

Résumé

Let $\Omega$ be a bounded open subset of $\Bbb R^n$, let $X=(x,t)$ be a point of $\Bbb R^n\times \Bbb R^N$. In the cylinder $Q=\Omega \times (-T,0)$, $T>0$, we deduce the local differentiability result $$ u \in L^2(-a,0,H^2(B(\sigma ),\Bbb R^N))\cap H^1(-a,0,L^2(B(\sigma ),\Bbb R^N)) $$ for the solutions $u$ of the class $L^q(-T,0,H^{1,q}(\Omega,\Bbb R^N))\cap C^{0,\lambda}(\bar Q,\Bbb R^N)$ ($0\lambda1$, $N$ integer $\ge1$) of the nonlinear parabolic system $$ -\sum_{i=1}^n D_i a^i (X,u,Du)+\dfrac {\partial u}{\partial t} = B^0(X,u,Du) $$ with quadratic growth and nonlinearity $q\ge 2$. This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions $u$ belonging to $W^{1,q}\cap C^{0,\lambda}$.

Détail

Classification : 35D10, 35K40, 35K55
Keywords: differentiability of weak solution; parabolic systems; nonlinearity with $q>2$