| Title:
|
On the fractional differentiability of the spatial derivatives of weak solutions to nonlinear parabolic systems of higher order (English) |
| Author:
|
Amato, Roberto |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
293-305 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We are concerned with the problem of differentiability of the derivatives of order $m+1$ of solutions to the “nonlinear basic systems” of the type $$ (-1)^m \sum _{|\alpha | = m}D^{\alpha } A^{\alpha }\big (D^{(m)}u\big )+ \frac {\partial u}{\partial t} = 0 \quad \text {in} \ Q. $$ We are able to show that $$ D^{\alpha }u \in L^2\bigl (-a, 0, H^{\vartheta }\big (B(\sigma ),\mathbb {R}^N\big )\big ), \quad |\alpha |=m+1, $$ for $\vartheta \in (0, {1}/{2})$ and this result suggests that more regularity is not expectable. (English) |
| Keyword:
|
nonlinear parabolic system |
| Keyword:
|
fractional differentiability |
| Keyword:
|
spatial derivative |
| Keyword:
|
weak solution |
| MSC:
|
35K41 |
| MSC:
|
35R11 |
| idZBL:
|
Zbl 06604467 |
| idMR:
|
MR3519602 |
| DOI:
|
10.1007/s10587-016-0256-z |
| . |
| Date available:
|
2016-06-16T12:36:14Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145723 |
| . |
| Reference:
|
[1] Amato, R.: Local differentiability for the solutions to basic systems of higher order.Matematiche 42 (1987), 109-119. Zbl 0693.35025, MR 1030910 |
| Reference:
|
[2] Campanato, S.: Elliptic Systems in Divergence Form. Interior Regularity.Quaderni, Scuola Normale Superiore, Pisa (1980), Italian. MR 0668196 |
| Reference:
|
[3] Campanato, S.: Sulla regolarità delle soluzioni di equazioni differenzialli di tipo ellittico.Editrice Tecnico Scientifica, Pisa (1963), Italian. |
| Reference:
|
[4] Fattorusso, L.: A result of differentiability of nonlinear parabolic systems under monotonicity hypothesis.Rend. Circ. Mat. Palermo, II. Ser. 39 (1990), 412-426 Italian. English summary. Zbl 0733.35026, MR 1119738 |
| Reference:
|
[5] Fattorusso, L.: Differentiability of solutions of nonlinear second order parabolic systems with quadratic behaviour.Boll. Unione Mat. Ital., VII. Ser. B 1 (1987), 741-764 Italian. English summary. Zbl 0656.35061, MR 0916291 |
| Reference:
|
[6] Fattorusso, L.: New contributions to the differentiability of weak solutions of nonlinear parabolic systems of order $2m$ with quadratic growth.Matematiche 41 (1986), 183-203 Italian. English summary. Zbl 0692.35024, MR 0998696 |
| Reference:
|
[7] Fattorusso, L.: On the differentiability of weak solutions of nonlinear second order parabolic equations with quadratic growth.Matematiche 40 (1985), 199-215 Italian. English summary. Zbl 0668.35045, MR 0959879 |
| Reference:
|
[8] Fattorusso, L., Marino, M.: Local differentiability of nonlinear parabolic systems of second order with nonlinearity $q\geq 2$.Ric. Mat. 41 (1992), 89-112 Italian. English summary. MR 1305346 |
| Reference:
|
[9] Marino, M., Maugeri, A.: Partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth.Rend. Semin. Mat. Univ. Padova 76 (1986), 219-245. Zbl 0622.35030, MR 0881572 |
| Reference:
|
[10] Naumann, J.: On the interior differentiability of weak solutions of parabolic systems with quadratic growth nonlinearities.Rend. Semin. Mat. Univ. Padova 83 (1990), 55-70. Zbl 0823.35027, MR 1066428 |
| . |
