Very weak solutions of higher-order degenerate parabolic systems. (English) Zbl 1178.35215

Bögelein, Verena

Adv. Differ. Equ. 14, No. 1-2, 121-200 (2009).

The author study the parabolic \(p\)-Laplacian system of order \(2m\), \(m \in \mathbb N\), in its weak formulation
\[ \int_{\Omega_T}u\cdot\varphi - \langle |D^mu|^{p-2}D^mu,D^m\varphi\rangle \,dz = 0,\quad \forall\varphi \in C_0^{\infty}(\Omega_T;\mathbb R^N). \]
The so-called very weak solution of the associated parabolic system is defined as a function
\[ u \in L^{p-\beta}(-T,0;W^{m,p-\beta}(\Omega;\mathbb R^N))\cap L^2(\Omega_T;\mathbb R^N) \]
satisfying the above-mentioned integral identity. The main result of the paper is to ensure the existence of an exponent \(\beta > 0\) depending only on the data such that any very weak solution \(u\) is a weak solution i.e. \(D^m u \in L^{p-\beta} \Rightarrow D^mu \in L^{p+\beta}\).

Reviewer: Vladimir N. Grebenev (Novosibirsk)

MSC:

35K65	Degenerate parabolic equations
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K59	Quasilinear parabolic equations
35D30	Weak solutions to PDEs
35B65	Smoothness and regularity of solutions to PDEs
35K41	Higher-order parabolic systems

Keywords:

very weak solution and weak solution; parabolic \(p\)-Laplacian