×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] H. Amann, Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal. 92 (1986), no. 2, 153-192. Zbl0596.35061 MR816618 · Zbl 0596.35061
[2] A. Arkhipova, Global solvability of the Cauchy-Dirichlet problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables, J. Math. Sci. 92 (1998), no. 6, 4231-4255. Zbl0953.35059 MR1668390 · Zbl 0953.35059
[3] A. Arkhipova, Local and global in time solvability of the Cauchy-Dirichlet problem to a class of nonlinear nondiagonal parabolic systems, St. Petersburg Math J. 11 (2000), no. 6, 81-119. Zbl0973.35095 MR1746069 · Zbl 0973.35095
[4] A. Arkhipova, Cauchy-Neumann problem to a class of nondiagonal parabolic systems with quadratic growth nonlinearities. I. Local and global solvability results, Comment. Math. Univ. Carolin. 41, (2000), no. 4, 693-718. Zbl1046.35047 MR1800172 · Zbl 1046.35047
[5] A. Arkhipova, Cauchy-Neumann problem to a class of nondiagonal parabolic systems with quadratic growth nonlinearities. II. Continuability of smooth solutions, to appear in Comment. Math. Univ. Carolin. (2000). Zbl1046.35047 MR1800172 · Zbl 1046.35047
[6] S. Campanato, Equazioni paraboliche del secondo ordine e spazi \({\mathcal{L}}^{2, \delta } (\Omega , \delta )\), Ann. Mat. Pura Appl., Ser. 4 73 (1966), 55-102. Zbl0144.14101 MR213737 · Zbl 0144.14101
[7] M. Giaquinta, “Multiple integrals in the calculus of variations and nonlinear elliptic systems”, Princeton, NJ, 1983. Zbl0516.49003 MR717034 · Zbl 0516.49003
[8] M. Giaquinta - G. Modica, Local existence for quasilinear parabolic systems under non-linear boundary conditions, Ann. Mat. Pura Appl. Ser. 4, \(\mathbf {149}\) (1987), 41-59. Zbl0655.35049 MR932775 · Zbl 0655.35049
[9] O. A. Ladyzhenskaja - V. A. Solonnikov - N. N. Uraltseva, “Linear and quasilinear equations of parabolic type”, Amer. Math. Soc. Providence, RI, 1968. · Zbl 0174.15403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.