Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
differentiability of weak solution; parabolic systems; nonlinearity with $q>2$
differentiability of weak solution; parabolic systems; nonlinearity with $q>2$
Summary:
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 $L^q(-T,0,H^{1,q}(\Omega,\Bbb R^N))\cap C^{0,\lambda}(\bar Q,\Bbb R^N)$ ($0<\lambda<1$, $N$ 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 $W^{1,q}\cap C^{0,\lambda}$.
References:
[1] Campanato S.: Sistemi ellittici in forma di divergenza. Regolarità all'interno. Quaderni Scuola Norm. Sup. Pisa, 1980. MR 0668196
[2] Campanato S.: Differentiability of the solutions of nonlinear elliptic systems with natural growth. Ann. Mat. Pura Appl. (4) 131 (1982). MR 0681558 | Zbl 0493.35022
[3] Fattorusso L.: Sulla differenziabilità delle soluzioni di sistemi parabolici non lineari del secondo ordine ad andamento quadratico. Boll. Un. Mat. Ital. B (7) 1 (1987), 741-764.
[4] Fattorusso L., Marino M.: Differenziabilità locale per sistemi parabolici non lineari del secondo ordine con non linearità $q\ge 2$. Ricerche Mat. 41 1 (1992), 89-112. MR 1305346
[5] Fattorusso L.: Differenziabilità locale per sistemi parabolici non lineari del secondo ordine con non linearità $1. Matematiche (Catania) 48 2 (1993), 331-347 (1994).
[6] Marino M., Maugeri M.: Differentiability of weak solutions of nonlinear parabolic systems with quadratic growth. Matematiche (Catania) 50 (1995), 2 361-377. MR 1414643 | Zbl 0907.35034
Search
Partner of
