Global Solvability of the Cauchy-Dirichlet Problem for a Class of Strongly Nonlinear Parabolic Systems
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
GLOBAL SOLVABILITY OF THE CAUCHY-DIRICHLET PROBLEM FOR A CLASS OF STRONGLY NONLINEAR PARABOLIC SYSTEMS A. A. Arkhipova St. Petersburg State University 28, Universitetskii pr., Petrodvorets, St. Petersburg 198504, Russia [email protected]
UDC 517.9
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. Bibliography: 15 titles.
1
Introduction
We consider a class of nonlinear parabolic systems with nondiagonal principal matrices. The elliptic operators of the systems are the Euler-Lagrange operators for a class of quadratic functionals. Additional terms in the systems admit quadratic growth in the gradient and arbitrary polynomial growth with respect to solutions. Under certain conditions on the data, we prove the time-global solvability of the Cauchy–Dirichlet problem in the case of two spatial variables. Our assumptions guarantee the H¨older continuity of weak solutions. Under additional stronger assumptions on the data, we prove the existence of the second order spatial derivatives of the solution. To describe the elliptic operator of the system, we fix a parameter t ∈ [0, ∞) and consider the functional {f (x, t, v, ∇ v) + g(x, t, v)} dx,
F t [v] =
x ∈ Ω ⊂ R2 ,
(1.1)
Ω
where Ω is a bounded domain with sufficiently smooth boundary ∂Ω, v(·, t) : Ω → RN , N > 1, kN . v = (v 1 , . . . , v N ), ∇ v = {vxkα }α2 We assume that the nonnegative function f (x, t, v, p) is smooth enough and admits quadratic growth in p ∈ R2N as |p| → ∞. We fix an arbitrary m ∈ N and assume that 0 g(x, t, v) b0 |v|m + ψ(x, t), b0 = const, and ψ is a given function. We describe the main assumptions in Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 19-44. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0201
201
Section 2 (cf. Assumptions A1 –A8 below). To prove the further smoothness of the solution, we additionally impose Assumption A9 . The Euler–Lagrange operator L = {Lk }kN for F [v] has the following structure: Lk [v] = −
d f k (x, t, v, ∇ v) + bk (x, t, v, ∇ v), d xα vxα
(1.2)
bk (x, t, v, p) = fvk (x, t, v, p) + gvk (x, t, v), |b(x, t, v, p)| a0 |p|2 + b0 |v|m−1 + ψ(x, t),
(1.3) a0 = const .
(1.4)
We study solutions u : Ω × [0, ∞) → RN , u = u(x, t), to the problem ut − div fp (x, t, u, ∇ u) + fu (x, t, u, ∇ u) + gu (x, t, u) = 0, u|Γ = 0,
Γ = ∂Ω × (0, ∞),
u|t=0 = ϕ0 (x),
x ∈ Ω.
(1.5)
The condition (1.4) means that the nonlinear parabolic systems can have strong (quadratic) nonlinearity with respect to the gradient and arbitrary polynomial growth with respect to the solution. The two-dimensional Ca
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