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Continuability in time of smooth solutions of strong–nonlinear nondiagonal parabolic systems. (English) Zbl 1049.35093
The author considers a strongly nonlinear parabolic system which has variational structure. Let $$\Omega\subseteq \mathbb{R}^n$$, $$n\geq 2$$ be a smooth bounded domain, set $$\mathcal{Q}_T=Q=\Omega\times (0,T)$$ with $$T>0$$ arbitrary. One considers a quadratic form (1) $2\,f(x,u,p)=\langle A(x,u)p,p\rangle=\sum A_{kj}^{\alpha\beta}(x,u)\,p_{\beta}^j\,p_{\alpha}^k,$ $$1\leq\alpha,\beta\leq n$$, $$1\leq k,j\leq N$$. The coefficients $$A_{kj}^{\alpha\beta}(x,u)$$ are smooth in $$x\in\mathbb{R}^n$$, $$u\in\mathbb{R}^N$$ and subject to certain conditions A1–A3, where, e.g. A2 requires (2) $\langle A(x,u)p,p\rangle\geq\nu\,| p| ^2, \text{for some $$\nu>0$$, $$p\in\mathbb{R}^{Nn}$$, $$x\in\mathbb{R}^n$$, $$u\in\mathbb{R}^N$$}.$ Setting $$E(u)=\int_{\Omega} f(x,u,u_x)\,dx$$, $$u_x=(\nabla u^1,\dots,\nabla u^N)$$, there is a system of Euler–Lagrange equations associated with $$E(u)$$, giving rise to the operators $$L^{(k)}$$ via $L^{(k)}u=-\left( A_{kj}^{\alpha\beta}(x,u)\,u_{x_{\beta}}^j\right)_{x_{\alpha}} +\tfrac{1}{2}\,\left(A_{mj}^{\alpha\beta}(x,u)\right)_{u^k}\,u_{x_{\beta}}^j\,u_{x_{\alpha}}^m$ (summation convention). The system under scrutiny is (3) $\partial_t u^k+L^{(k)} u=0,\quad (x,t)\in Q,\quad u| _{\gamma}=0, u| _{t=0}=\phi, \quad \gamma=\times (0,T).$ The system (3) is cast in a functional setting which can only be indicated. $$B_R(x_0)\subseteq\mathbb{R}^n$$ is the open sphere centered at $$x_0$$; set $$\Omega_R(x_0)=B_R(x_0)\cap\Omega$$ and $$Q_R(z_0)=\Omega_R(x_0)\times\Lambda_R(t_0)$$, where $$\Lambda_R(t_0)=(t_0-R^2,t_0)$$. The functional setting is essentially on Hölder spaces, among which $$H^{2+\alpha,1+\alpha/2}(\overline{Q})$$ is a standard space, adapted to the parabolic setting. Morrey spaces $$L^{p,\alpha}(Q,\delta)$$ and Campanato spaces also play a role. Finally the space of solution candidates of (3) is defined as follows: $$K_{\alpha}\{[t_1,t_2]\}$$ is the set of $$v:\overline{Q'}\to\mathbb{R}^N$$ such that $$v\in H^{2+\alpha,1+\alpha/2}(\overline{Q'})$$, $$v_{xt}\in L^{2,n+2\alpha}(Q';\delta)$$ where $$Q'=\Omega\times (t_1,t_2)$$, $$t_1,t_2\in[ 0,T]$$, $$\alpha\in( 0,1)$$. Finally one sets $$K_{\alpha}\{[t_1,t_2)\}=\cap K_{\alpha}\{[t_1,\tau]\}$$, $$\tau\in (t_1,t_2)$$. The first main result, Theorem 1, then is as follows. Assume A1–A3, let $$\partial\Omega\in C^{2+\alpha}$$, $$\phi\in C^{2+\alpha}(\overline{\Omega})$$ (for some $$\alpha\in (0,1)$$). Let $$u\in K_{\alpha}\{[0,T)\}$$ be a solution of (3). Then there is $$\varepsilon_0$$ such that: if for some $$R_0$$, (4) $\sup_{t_0,x_0}\,\sup_{\rho}\,\rho^{1/n}\,\int_{Q_{\rho}(z_0)} | u_x| ^2\,dz<\varepsilon_0,\quad t_0\in\left[ T/2,T\right),\quad x_0\in\overline{\Omega},$ where $$\rho\in (0,R_0]$$, $$z_0=(x_0,t_0)$$, $$dz=dx\,dt$$, then $$u\in K_{\alpha}\{[0,T]\}$$. The number $$\epsilon_0$$ depends only on the constants in A2 and A3. The proof splits into several lemmas, the first of which, based on the variational structure, asserts some inequalities. The further lemmas prepare the ground for for inequality (4). There is a Theorem 2 which, among other things, states that, if $$u\in K_{\alpha}\{[0,T]\}$$ is a maximal solution of (3), then the singular set $$\sigma\times\{T\}$$ of $$u$$ has Hausdorff dimension $$R_{n-2}(\sigma)\leq c_0$$, with $$c_0$$ depending on the same constants as $$\epsilon_0$$ in Theorem 1.

##### MSC:
 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35K45 Initial value problems for second-order parabolic systems 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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