| Title:
|
Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity $q\ge 2$ (English) |
| Author:
|
Fattorusso, Luisa |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
73-90 |
| . |
| Category:
|
math |
| . |
| 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 $W^{1,q}\cap C^{0,\lambda}$. (English) |
| Keyword:
|
differentiability of weak solution |
| Keyword:
|
parabolic systems |
| Keyword:
|
nonlinearity with $q>2$ |
| MSC:
|
35D10 |
| MSC:
|
35K40 |
| MSC:
|
35K55 |
| idZBL:
|
Zbl 1098.35054 |
| idMR:
|
MR2076860 |
| . |
| Date available:
|
2009-05-05T16:43:21Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119437 |
| . |
| Reference:
|
[1] Campanato S.: Sistemi ellittici in forma di divergenza. Regolarità all'interno.Quaderni Scuola Norm. Sup. Pisa, 1980. MR 0668196 |
| Reference:
|
[2] Campanato S.: Differentiability of the solutions of nonlinear elliptic systems with natural growth.Ann. Mat. Pura Appl. (4) 131 (1982). Zbl 0493.35022, MR 0681558 |
| Reference:
|
[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. |
| Reference:
|
[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 |
| Reference:
|
[5] Fattorusso L.: Differenziabilità locale per sistemi parabolici non lineari del secondo ordine con non linearità $1<q<2$.Matematiche (Catania) 48 2 (1993), 331-347 (1994). |
| Reference:
|
[6] Marino M., Maugeri M.: Differentiability of weak solutions of nonlinear parabolic systems with quadratic growth.Matematiche (Catania) 50 (1995), 2 361-377. Zbl 0907.35034, MR 1414643 |
| . |
