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

Comment.Math.Univ.Carolinae 45,1 (2004) 73-90.

Abstract: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 $L^q(-T,0,H^{1,q}(\Omega ,\Bbb R^N))\cap C^{0,\lambda }(\mathaccent "7016 Q,\Bbb R^N)$ ($0<\lambda <1$, $N$ integer $\ge 1$) of the nonlinear parabolic system $$ -\sum _{i=1}^n D_i a^i (X,u,Du)+{\displaystyle {\partial u\over \partial t}} = B^0(X,u,Du) $$ 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 $W^{1,q}\cap C^{0,\lambda }$.

Keywords: differentiability of weak solution, parabolic systems, nonlinearity with $q>2$
AMS Subject Classification: 35K55