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

Fattorusso, Luisa. "Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity $q\ge 2$." Commentationes Mathematicae Universitatis Carolinae 45.1 (2004): 73-90. <http://eudml.org/doc/249350>.

@article{Fattorusso2004,
abstract = {Let $\Omega $ be a bounded open subset of $\mathbb \{R\}^n$, let $X=(x,t)$ be a point of $\mathbb \{R\}^n\times \mathbb \{R\}^N$. In the cylinder $Q=\Omega \times (-T,0)$, $T>0$, we deduce the local differentiability result \[ u \in L^2(-a,0,H^2(B(\sigma ),\mathbb \{R\}^N))\cap H^1(-a,0,L^2(B(\sigma ),\mathbb \{R\}^N)) \] for the solutions $u$ of the class $L^q(-T,0,H^\{1,q\}(\Omega ,\mathbb \{R\}^N))\cap C^\{0,\lambda \}(\bar\{Q\},\mathbb \{R\}^N)$ ($0<\lambda <1$, $N$ integer $\ge 1$) of the nonlinear parabolic system \[ -\sum \_\{i=1\}^n D\_i a^i (X,u,Du)+\dfrac\{\partial u\}\{\partial t\} = B^0(X,u,Du) \] 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 $W^\{1,q\}\cap C^\{0,\lambda \}$.},
author = {Fattorusso, Luisa},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {differentiability of weak solution; parabolic systems; nonlinearity with $q>2$; differentiability of weak solution; parabolic systems; nonlinearity with ; local differentiability; interpolation theory; imbedding theorem of Gagliardo-Nirenberg type},
language = {eng},
number = {1},
pages = {73-90},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity $q\ge 2$},
url = {http://eudml.org/doc/249350},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Fattorusso, Luisa
TI - Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity $q\ge 2$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 1
SP - 73
EP - 90
AB - Let $\Omega $ be a bounded open subset of $\mathbb {R}^n$, let $X=(x,t)$ be a point of $\mathbb {R}^n\times \mathbb {R}^N$. In the cylinder $Q=\Omega \times (-T,0)$, $T>0$, we deduce the local differentiability result \[ u \in L^2(-a,0,H^2(B(\sigma ),\mathbb {R}^N))\cap H^1(-a,0,L^2(B(\sigma ),\mathbb {R}^N)) \] for the solutions $u$ of the class $L^q(-T,0,H^{1,q}(\Omega ,\mathbb {R}^N))\cap C^{0,\lambda }(\bar{Q},\mathbb {R}^N)$ ($0<\lambda <1$, $N$ integer $\ge 1$) of the nonlinear parabolic system \[ -\sum _{i=1}^n D_i a^i (X,u,Du)+\dfrac{\partial u}{\partial t} = B^0(X,u,Du) \] 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 $W^{1,q}\cap C^{0,\lambda }$.
LA - eng
KW - differentiability of weak solution; parabolic systems; nonlinearity with $q>2$; differentiability of weak solution; parabolic systems; nonlinearity with ; local differentiability; interpolation theory; imbedding theorem of Gagliardo-Nirenberg type
UR - http://eudml.org/doc/249350
ER -