## Advances in Differential Equations

- Adv. Differential Equations
- Volume 14, Number 1/2 (2009), 121-200.

### Very weak solutions of higher-order degenerate parabolic systems

#### Abstract

We consider non-linear higher-order parabolic systems whose simplest model is the parabolic $p$-Laplacean system \begin{equation*} \int_{\Omega_T} u\cdot \varphi_t - \langle |D^mu|^{p-2}D^mu,D^m\varphi\rangle \,dz = 0. \end{equation*} It turns out that the usual regularity assumptions on solutions can be weakened in the sense that going slightly below the natural integrability exponent still yields a classical weak solution. Namely, we prove the existence of some $\beta>0$ such that $D^mu\in L^{p-\beta} \Rightarrow D^mu\in L^{p+\beta}$.

#### Article information

**Source**

Adv. Differential Equations Volume 14, Number 1/2 (2009), 121-200.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867280

**Mathematical Reviews number (MathSciNet)**

MR2478931

**Zentralblatt MATH identifier**

1178.35215

**Subjects**

Primary: 35D10 35G20: Nonlinear higher-order equations 35K65: Degenerate parabolic equations

#### Citation

Bögelein, Verena. Very weak solutions of higher-order degenerate parabolic systems. Adv. Differential Equations 14 (2009), no. 1/2, 121--200. https://projecteuclid.org/euclid.ade/1355867280