Geodesic
On the fractional differentiability of the spatial derivatives of weak solutions to nonlinear parabolic systems of higher order
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 293-305
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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.
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.
DOI : 10.1007/s10587-016-0256-z
Classification : 35K41, 35R11
Keywords: nonlinear parabolic system; fractional differentiability; spatial derivative; weak solution 
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Amato, Roberto. On the fractional differentiability of the spatial derivatives of weak solutions to nonlinear parabolic systems of higher order. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 293-305. doi: 10.1007/s10587-016-0256-z

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