Influence of variable coefficients on global existence of solutions of semilinear heat equations with nonlinear boundary conditions

We consider semilinear parabolic equations with nonlinear boundary conditions. We give conditions which guarantee global existence of solutions as well as blow-up in finite time of all solutions with nontrivial initial data. The results depend on the behavior of variable coefficients as $t \to \infty.$

Blow-up problem for parabolic equations with reaction term in general form were considered in many papers (see, for example, [1] - [7] and the references therein). For the global existence and blow-up of solutions for linear parabolic equations with β(t) ≡ 1 in (1.2), we refer to previous studies [8] - [13]. In particular, Walter [9] proved that if g(s) and g ′ (s) are continuous, positive and increasing for large s, a necessary and sufficient condition for global existence is +∞ ds g(s)g ′ (s) = +∞.
Some papers are devoted to blow-up phenomena in parabolic problems with timedependent coefficients (see, for example, [14] - [20]). So, it follows from results of Payne and Philippin [15] blow-up of all nontrivial solutions for (1.1)-(1.3) with β(t) ≡ 0 under the conditions (2.15) and where z satisfies +∞ a ds z(s) < +∞ for any a > 0 and Jensen's inequality 1 In (1.4), |Ω| is the volume of Ω. The aim of our paper is study the influence of variable coefficients α(t) and β(t) on the global existence and blow-up of classical solutions of (1.1)-(1.3).
This paper is organized as follows. Finite time blow-up of all nontrivial solutions is proved in Section 2. In Section 3, we present the global existence of solutions for small initial data.

Finite time blow-up
In this section, we give conditions for blow-up in finite time of all nontrivial solutions of (1.1) -(1.3).
Before giving our main results, we state a comparison principle which has been proved in [21], [22] for more general problems.
The first our blow-up result is the following. Proof. We suppose that u(x, t) is a nontrivial nonnegative solution which exists in Q T for any positive T. Then for some According to Theorem 3.6 of [23] it yields ∂u(x ⋆ , t ⋆ )/∂ν < 0, which contradicts the boundary where σ is a positive constant.
Let G N (x, y; t − τ ) denote the Green's function for the heat equation given by with homogeneous Neumann boundary condition. We note that the Green's function has the following properties (see, for example, [24], [25]): for some small ε > 0. Here by c i (i ∈ N) we denote positive constants. Now we introduce several auxiliary functions. We suppose that h(s) and γ(t) be a positive continuous function for t ≥ t 0 such that γ(t 0 ) = β(t 0 )h(2σ) and We consider the following problem To find lower bound for v(x, t) we represent (2.9) (2.10) Using (2.7), (2.8) and the properties of the Green's function (2.4), (2.5), we obtain As in [13] we put We observe that m(t) is well defined and positive for t ≥ t 0 .
Since v(x, t) is the solution of (2.9), we get Applying the inequality h ′ (v) ≥ 0, Gauss theorem, the boundary condition in (2.9) and (2.11), we obtain for t ≥ t 0 .
Since m(t) > 0 and ε, ξ(t), γ(t) are arbitrary we conclude that the solution of (1. To prove next blow-up result for (1.1) -(1.3) we need a comparison principle with unstrict inequality in the boundary condition.
For positive ε we introduce Passing to the limits as ε → 0 and τ → T, we prove the theorem. We consider the following auxiliary problem We prove at first that v(t) > σ for t ≥ t 0 .
Remark 3.2. We suppose that g(s) is a nondecreasing positive function for s > 0, f (s) > 0 for s > 0 and (2.1), (2.14) hold. Then by Theorem 2.2 and Theorem 2.6 (3.4) is necessary for global existence of solutions of (1.1)-(1.3). Let for any a > 0 g(s) > δ(a) > 0 if s > a. Then arguing in the same way as in the proof of Lemma 3.3 of [28] it is easy to show that (3.5) is necessary for the existence of nontrivial bounded global solutions of (1.1)-(1.3).