Blow-Up Profile of Solutions in Parabolic Equations with Nonlocal Dirichlet Conditions
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Blow-Up Profile of Solutions in Parabolic Equations with Nonlocal Dirichlet Conditions Bingchen Liu1
· Changcheng Zhang1 · Yu Wei1
Received: 21 January 2019 / Accepted: 28 November 2019 © Iranian Mathematical Society 2019
Abstract This paper deals with parabolic equations with nonlocal Dirichlet boundary conditions. Critical exponent and simultaneous blow-up criteria are studied for blow-up solutions. Moreover, we determine uniform blow-up profiles and blow-up set to all kinds of simultaneous blow-up solutions. It is interesting that large weighted functions could provide sufficient help to the occurring of blow-up of solutions without conditions on initial data. Keywords Nonlocal boundary condition · Simultaneous blow-up · Uniform blow-up profile Mathematics Subject Classification 35K65 · 35K61 · 35B33 · 35B40
1 Introduction and Main Results In this paper, we study the following parabolic equations coupled via nonlinear nonlocal sources:
Communicated by Ali Abkar.
B
Bingchen Liu [email protected] Changcheng Zhang [email protected] Yu Wei [email protected]
1
College of Science, China University of Petroleum, Qingdao 266555, Shandong Province, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
⎧ p1 ⎪ m n ⎪ u (x, t) = u(x, t) + u (x, t)v (x, t)dx , (x, t) ∈ × (0, T ), t ⎪ ⎪ ⎪ ⎪ q1 ⎪ ⎪ ⎪ ⎪ v (x, t) = v(x, t) + u p (x, t)v q (x, t)dx , (x, t) ∈ × (0, T ), ⎪ ⎨ t (1.1) u(x, t) = ϕ(x, y)u(y, t)dy, v(x, t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ψ(x, y)v(y, t)dy, (x, t) ∈ ∂ × (0, T ), ⎪ ⎪ ⎪ ⎩ u(x, 0) = u 0 (x), v(x, 0) = v0 (x), x ∈ , where ⊂ R N is a bounded domain with a smooth boundary ∂; T denotes the maximal existence time of system (1.1); Exponents m, n, p, q, p1 , q1 are nonnegative constants satisfying that m + n > 0, p + q > 0, and p1 , q1 > 0; Weighted functions ¯ [0, +∞)). Assume that u 0 (x), v0 (x) ∈ C 2,ν (, R + ) ϕ(x, y), ψ(x, y) ∈ C(∂ × , with constant 0 < ν < 1, satisfying the compatible conditions ϕ(x, y)u 0 (y)dy, v0 (x) = ψ(x, y)v0 (y)dy, x ∈ ∂. u 0 (x) =
Using the methods given in the works [3,8], we know that system (1.1) has local nonnegative classical solutions and the classical comparison principle holds also. Moreover, if the nonlinear sources are locally Lipschitz continuous in u and v, the uniqueness of the solutions holds. Nonlinear parabolic equations coupled via nonlocal sources in (1.1) are widely used in population dynamics, chemical reactions, and heat transfer, etc., where u and v represent the densities of two biological populations during a migration, the thickness of two kinds of chemical reactants, and the temperatures of two different materials during a propagation, etc. For the problem (1.1) with p1 = q1 = 1 and ϕ = ψ ≡ 0, Li, Huang and Xie in [4] discussed the blow-up criteria and uniform blow-up profiles of solutions. For system (1.1) with p1 = q1 = 1, Kong and Wang in [3] show the blow-up criteria, simultaneous blow-up criteria, blow-up sets, and uniform blow-up profiles for the solutions. As for some general
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