\documentclass{article} \usepackage{amsmath,amssymb} \begin{document} Let $\Omega$ be a bounded open subset of $R^{n}$, let $X = (x, t)$ be a point of $R^{n}\times R^{N}$ In the cylinder $Q = \Omega \times(-T, 0)$,$T > 0$, we deduce the local differentiability result $u \in L^{2}(-a,0,H^{2}(B(\sigma),\mathbb{R}^{N}))\cap H^{1}(-a,o,L^{2}(B(\sigma),\mathbb{R}^{N}))$ for the solutions u of the class $L^q(-T, 0,H^{1,q}(\Omega,R^{N})) \cap C^{0,\lambda}(\overline Q,R^{N}))$ $(0 < \lambda < 1, N$ integer $\geq 1)$ of the nonlinear parabolic system $$-\sum_{j=1}^{n}D_{j}a^{j}(X,u,Du)+\frac{\partial u}{\partial t}= B^{0}(X,u,Du)$$ with quadratic growth and nonlinearity $q \geq 2$. This result had been obtained making use of the interpolation theory and an embedding theorem of Gagliardo-Nirenberg type for functions u belonging to $W^{1,q}\cap C^{0,\lambda}$. \end{document}

### Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and non linearity $qge2$

#### Abstract

\documentclass{article} \usepackage{amsmath,amssymb} \begin{document} Let $\Omega$ be a bounded open subset of $R^{n}$, let $X = (x, t)$ be a point of $R^{n}\times R^{N}$ In the cylinder $Q = \Omega \times(-T, 0)$,$T > 0$, we deduce the local differentiability result $u \in L^{2}(-a,0,H^{2}(B(\sigma),\mathbb{R}^{N}))\cap H^{1}(-a,o,L^{2}(B(\sigma),\mathbb{R}^{N}))$ for the solutions u of the class $L^q(-T, 0,H^{1,q}(\Omega,R^{N})) \cap C^{0,\lambda}(\overline Q,R^{N}))$ $(0 < \lambda < 1, N$ integer $\geq 1)$ of the nonlinear parabolic system $$-\sum_{j=1}^{n}D_{j}a^{j}(X,u,Du)+\frac{\partial u}{\partial t}= B^{0}(X,u,Du)$$ with quadratic growth and nonlinearity $q \geq 2$. This result had been obtained making use of the interpolation theory and an embedding theorem of Gagliardo-Nirenberg type for functions u belonging to $W^{1,q}\cap C^{0,\lambda}$. \end{document}
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differentiability of weak solutions; parabolic systems; quadratic growth
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12318/8554
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