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On classical solvability of the Cauchy-Dirichlet problem for nondiagonal parabolic systems in the case of two spatial variables
Abstract
We consider a class of nondiagonal nonlinear parabolic systems with strong nonlinearities in the gradient and prove almost everywhere smooth solvability of the Cauchy-Dirichlet problem.
... Global weak solvability of the Cauchy-Dirichlet and Cauchy-Neumann problems for strongly nonlinear nondiagonal systems (arising as variational flows) was proved in the case of two spatial variables [13][14][15]. Constructed in these works global solutions are smooth almost everywhere on × [0, ∞) and may have at most finitely many singular points. Recently, it was stated that the singular points in such problem are absent if the one-side condition on the nonlinear term is assumed [16], or if another restriction on b 0 in (3) is prescribed [17]. ...
... for a sequence m → ∞. Moreover, by (15), |u 0 | ≤ M. Then J m =: ...
We consider quasilinear parabolic systems of equations with nondiagonal principal matrices and quadratic nonlinearities in the gradient. Under a one-side condition on the nonlinear terms, we study the local smoothness of the possibly unbounded weak solutions.
... But our work is different in that the principal matrices of the systems (1.5) are nondiagonal and more general operators are considered. The result of [1] was generalized in [4]- [6]. Parabolic systems of variational structure (g = 0 and n = 2) were considered in [7,8], where the existence of a global Hölder continuous weak solution was established under the condition b(x, t, u, p) · u −K, (x, t, u, p) ∈ Ω × (0, T ) × R N × R 2N , K = const 0. ...
... A 6 The following inequalities hold: ...
We consider a class of nonlinear parabolic systems for elliptic operators of variational structure with nondiagonal principal matrices. Additional terms in the systems can have quadratic growth with respect to the gradient and arbitrary polynomial growth with respect to solutions. We obtain sufficient conditions for the time-global weak solvability of the Cauchy–Dirichlet problem and study the regularity of the solution. The case of two spatial variables is considered.
... It cannot be expected that it is possible in all circumstances, as certain counterexamples seen to indicate that solutions may start smoothly and even remaind bounded, but develop a singularity after finite time, see [18,17], and also [12], [13]. Although, in some papers, for instance in [4], [6], [8], [14] and [15] the global existence result is proved under some structural conditions. This information leads to the possibility to control some lower-order norms "a-priory". ...
In this paper we study the quasilinear nondiagonal parabolic type systems. We
assume that the principal elliptic operator, which is part of the parabolic
system, has a divergence structure. Under certain conditions it is proved the
well-posedness of classical solutions, which exist globally in time.
The Cauchy–Dirichlet problem for the class of nondiagonal q-nonlinear parabolic systems is studied, 1 q < 2.="" in="" the="" case="" of="" two="" spatial="" variables,="" a="" solution="" that="" is="" global="" in="" time="" and="" smooth="" almost="" everywhere="" is="" constructed.="" hausdorff''s="" dimension="" of="" the="" singular="" set="" is="" estimated.="" bibliography:="" 16="">
We consider q-nonlinear nondiagonal elliptic systems, where 1 < q < 2, with strong nonlinear terms in the gradient. Under
a smallness condition on the gradient of a solution in the Morrey space Lq,n−q, we estimate the Lp-norm of the gradient for p > q and the Holder norm of the solution for the case n = 2. An abstract theorem on “quasireverse
Holder inequalities” proved by the author earlier is used essentially. Bibliography: 24 titles.
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