Blow-up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition

Global Existence and Blow-Up of Solutions to a Parabolic Nonlocal Equation Arising in a Theory of Thermal Explosion

“…(H3) There exists a constant γ > 0 such that inf gðuÞ ≥ γ Let function UðtÞ = sup x∈ Ω juðx, tÞj, where uðx, tÞ is a blow-up solution to equation (1). The following lemma is given according to [8,30,31]. Lemma 9.…”

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

“…In particular, the blow-up problem for nonlocal parabolic equations with boundary condition (2) was investigated in literature. [25][26][27][28][29][30][31][32] So, for example, Cui et al 25 studied (1)-(3) with b(x, t) ≡ 0, a(x, t) ≡ a(x) and k(x, , t) ≡ k(x, ), and problem (1)-(3) with r = 0, a(x, t) ≡ 1, b(x, t) ≡ b > 0 and k(x, , t) ≡ k(x, ) was considered in Mu et al 30 Gladkov and Guedda 8 studied (1)-(3) with a(x, t) ≡ 0. The existence of classical local solutions and the comparison principle for (1)-(3) were proved in Gladkov and Kavitova.…”

“…Initial-boundary value problems for nonlocal parabolic equations with nonlocal boundary conditions were considered in many papers also (see, for example, [4,7,9,10,26,27,34]). In particular, blow-up problem for nonlocal parabolic equations with boundary condition (1.2) was investigated in [5,8,25,28,29,31,35,36]. So, for example, the authors of [5] studied (1.1)-(1.3) with b(x, t) ≡ 0, a(x, t) ≡ a(x) and k(x, y, t) ≡ k(x, y), and problem (1.1)-(1.3) with r = 0, a(x, t) ≡ 1, b(x, t) ≡ b > 0 and k(x, y, t) ≡ k(x, y) was considered in [31].…”

“…In particular, blow-up problem for nonlocal parabolic equations with boundary condition (1.2) was investigated in [5,8,25,28,29,31,35,36]. So, for example, the authors of [5] studied (1.1)-(1.3) with b(x, t) ≡ 0, a(x, t) ≡ a(x) and k(x, y, t) ≡ k(x, y), and problem (1.1)-(1.3) with r = 0, a(x, t) ≡ 1, b(x, t) ≡ b > 0 and k(x, y, t) ≡ k(x, y) was considered in [31]. The authors of [15] studied (1.1)-(1.3) with a(x, t) ≡ 0.…”