Existence and Uniqueness of Renormalized Solutions to Parabolic Problems for Equations with Diffuse Measure
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Journal of Mathematical Sciences, Vol. 247, No. 6, June, 2020
EXISTENCE AND UNIQUENESS OF RENORMALIZED SOLUTIONS TO PARABOLIC PROBLEMS FOR EQUATIONS WITH DIFFUSE MEASURE F. Kh. Mukminov Institute of Mathematics, UFRC RAS 112, Chernyshevsky St., Ufa 450008, Russia M. Akmullah Bashkir State Pedagogical University 3A, Oktyabrskoi Revolyutsii St., Ufa 450000, Russia [email protected]
UDC 517.9
We consider the first mixed problem for anisotropic parabolic equations with variable nonlinearity exponents and diffuse measure on the right-hand side in the cylindrical domain (0, T ) × Ω, where Ω is a bounded domain. We establish the existence of a renormalized solution. The uniqueness of the renormalized solution is proved under the assumption that the functions entering the equation are locally Lipschitz. Bibliography: 25 titles.
1
Introduction
Let Ω be an arbitrary bounded domain in Rn = {x = (x1 , x2 , . . . , xn )}, n 1. In the cylindrical domain D T = (0, T ) × Ω, we consider the first mixed problem ut − div (a(t, x, u, ∇u) + Φ(t, x, u)) − b(t, x, u, ∇u) = μ, u(t, x) = 0, S = {t > 0} × ∂Ω,
a = (a1 , . . . , an ),
S
u(0, x) = u0 (x) ∈ L1 (Ω).
(1.1) (1.2) (1.3)
For a model example of Equation (1.1) with variable nonlinearity exponents we can consider the equation n (|uxi |pi (x)−2 uxi + bi (t, x))xi − F (t, x) = μ, ut − i=1
where μ = f + div G + gt in D , f, F ∈ L1 (D T ),
g|t=0 = 0,
bi , Gi ∈ Lpi (·) (D T ),
(1.4) ˚p0,1 (D T ), g∈W
Translated from Problemy Matematicheskogo Analiza 102, 2020, pp. 137-152. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2476-0900
900
(1.5)
p = (p1 (x), p2 (x), . . . , pn (x)), 0 < p− pi (x) p+ , x ∈ Ω, i = 1, n. The spaces Lpi (·) (D T ), −1 ˚p0,1 (D T ) are defined in Section 2. In the case pi (x) = p = const, i = 1, n, we = 1, W p−1 i + pi ˚p0,1 (D T ) spaces. The equality (1.4) is understood deal with the Lebesgue Lp (D T ) and Sobolev W in the sense of the theory of distributions. Moreover, it is not assumed that the distribution μ ∈ D generates a measure in D T . The equality g|t=0 = 0 in (1.4) is understood in the weak sense. Equations with measure are mainly of purely mathematical interest, but there are some physical applications of such equations. Thus, in [1], there is an example of the equation in R3 with the Dirac δ-function + 3/2
− Δu + 4π((u − λ) )
=
N
mi δai ,
i=1
u(x) → 0,
|x| → ∞,
which appears in the Thomas–Fermi theory on atoms and molecules. It is also required to determine the number λ 0 in the condition ((u − λ)+ )3/2 dx = I. As known, some difficulties can arise in establishing the existence and uniqueness of solutions to equations with measure. In particular, for the equation − div a(x, ∇u) = μ with a monotone flow a such that |a(x, y)| C(F (x)1−1/p + |y|p−1 ), the Dirichlet problem ˚q1 (Ω) u∈ W
n(p − 1) ∀ q ∈ 1, , n−1
a(x, y) · y α|y|p ,
a(x, ∇u)∇vdx = Ω
y ∈ Rn , F ∈ L1 (Ω),
vdμ,
v ∈ C0∞ (Ω),
(1.6)
Ω
with a bounded Radon measure μ was considered in [2], where the existence of a solution was p
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