Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition

In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.

In this paper we obtain necessary and sufficient conditions for the existence of global solutions as well as for a blow-up in finite time of solutions for problem (1.1)-(1.3). Our global existence and blow-up results depend on the behavior of the functions c(x, t) and k(x, y, t) as t → ∞.
This paper is organized as follows. The global existence theorem for any initial data and blow-up in finite time of solutions for large initial data are proved in section 2. In section 3 we present finite time blow-up of all nontrivial solutions as well as the existence of global solutions for small initial data. Finally, in section 4 we show that under certain conditions blow-up occurs only on the boundary.
Definition 2.1. We say that a nonnegative function u(x, t) ∈ C 2,1 (Q T )∩C 1,0 (Q T ∪ Γ T ) is a supersolution of (1.1)- (1.3) ∂u(x, t) ∂ν ≥ Ω k(x, y, t)u l (y, t) dy, x ∈ ∂Ω, 0 ≤ t < T, 3) and u(x, t) ∈ C 2,1 (Q T )∩C 1,0 (Q T ∪Γ T ) is a subsolution of (1. To prove the main results we use the positiveness of a solution and the comparison principle which have been proved in [21]. Theorem 2.3. Let u and u be a supersolution and a subsolution of problem (1.1)- The proof of a global existence result relies on the continuation principle and the construction of a supersolution. We suppose that Proof. In order to prove the existence of global solutions we construct a suitable explicit supersolution of (1.1)-(1.3) in Q T for any positive T. Suppose at first that l ≤ 1. Since k(x, y, t) is a continuous function there exists a constant K > 0 such that k(x, y, t) ≤ K (2.5) in ∂Ω × Q T . Let λ 1 be the first eigenvalue of the following problem and ϕ(x) be the corresponding eigenfunction with sup Ω ϕ(x) = 1. It is well known where constants C, µ and a are chosen to satisfy the following inequalities: Indeed, it is easy to check that for (x, t) ∈ S T and u(x, 0) ≥ u 0 (x) (2.8) for x ∈ Ω. It follows from (2.6)-(2.8) that problem (1.1)-(1.3) has a global solution for any initial datum. Suppose now that 1 < l < p and (2.4) holds. By (2.4) we have c(x, t) ≥ c in Q T , where c is some positive constant.
To construct a supersolution we use the change of variables in a neighborhood of ∂Ω as in [22]. Let x be a point in ∂Ω. We denote by n(x) the inner unit normal to ∂Ω at the point x. Since ∂Ω is smooth it is well known that there exists δ > 0 such that the mapping ψ : ∂Ω × [0, δ] → R n given by ψ(x, s) = x + s n(x) defines new coordinates (x, s) in a neighborhood of ∂Ω in Ω. A straightforward computation shows that, in these coordinates, ∆ applied to a function g(x, s) = g(s), which is independent of the variable x, evaluated at a point (x, s) is given by where H j (x) for j = 1, ..., n − 1, denotes the principal curvatures of ∂Ω at x.
for a suitable choice of ε. To estimate the integral I in the right hand side of (2.11) we shall use the change of variables in a neighborhood of ∂Ω. Let where J(y, s) is Jacobian of the change of variables. Then we have the inequality (2.11) holds if ε is small enough and hence by Theorem 2.3 we get Note that under β = 2/(l − 1) and a suitable choice of α in (2.10) the same proof holds if l = p > 1 and λ is large enough and consequently a solution of problem Now we shall prove finite time blow-up result. We suppose that Proof. At first we suppose that p ≤ 1, l > 1 and (2.12) holds with t 0 ≥ 0. To prove the theorem we construct a subsolution of an auxiliary problem which blows up in finite time. First of all we get a lower bound for solutions of (1.1)-(1.3) with positive initial data. We denote It is not difficult to check that Then by Theorem 2.3 we have Consider the change of variables in a neighborhood of ∂Ω as in Theorem 2.4. Set for some positive k 1 , γ and t 1 > t 0 .
Let us consider the following initial boundary value problem: (t 0 , t 1 ) and will be chosen later. We can define the notions of a supersolution and a subsolution of (2.17) in a similar way as in Definition 2.1. We shall use a comparison principle for a subsolution and a supersolution of (2.17) which can be proved analogously to Theorem 2.3. It is easy to see that u(x, t) is a supersolution of (2.17) in Q(γ, t 0 , where σ > 2/(l − 1) and show that ψ(s, t) is a subsolution of (2.17) in Q(γ, t 0 , t 2 ) under suitable choice of t 2 and γ. It is obvious, ψ(0, t) → ∞ as t → t 2 .
For 0 < s < γ and small γ we have To do this, we use the change of variables in a neighborhood of ∂Ω. Let where J(y, s) is Jacobian of the change of variables. By virtue of (2.16), We suppose now that Comparing u(x, t) and ψ(s, t) in Q(γ, t 0 , t 2 ) we prove the theorem for p ≤ 1, l > 1.
Let l > p > 1 and (2.12) hold with t 0 = 0. We denote c 1 = sup Qt 1 c(x, t) and suppose that where t 2 ∈ (0, t 1 ) and will be chosen later. It is not difficult to check that is a subsolution of (1.1)-(1.3) in Q t2 . Then by Theorem 2.3 we have In the same way as in a previous case we can show that ψ(s, t) is a subsolution of (2.17) in Q(γ, t 0 , t 2 ) with t 0 = 0 for small values of γ and Remark 2.7. We put Proof. Thanks to the assumptions of the theorem we have c(x, t) ≥ c 0 and k(x, y, t) ≤ K in Q τ and ∂Ω × Q τ , respectively, where c 0 , K and τ are some positive constants. Let ψ(x) be a positive solution of the following problem and suppose that f (t) is a solution of the following equation Then f (t) can be written in an explicit form We assume that Then f (t) ≡ 0 for t ≥ τ.
To prove the theorem we construct a supersolution of (1. for small values of f (0). By 3) for small initial data.
The following two statements deal with the case p = 1, l > 1. Let us introduce the notations where c(t) was defined in (2.13).
We prove that any nontrivial solution of (1.  and there exist positive constants α, t 0 and K such that α > t 0 and .

Proof.
Let v(x, t) be a solution of the following problem v t = ∆v for x ∈ Ω, t > 0, (3.9) ∂v(x, t) ∂ν = k c (x, t) Ω v l (y, t) dy for x ∈ ∂Ω, t > 0, (3.10) v(x, 0) = u 0 (x) for x ∈ Ω, (3.11) By a direct computation we can check that is a subsolution of (1.1)-(1.3) in Q T for any T > 0. Then by Theorem 2.3 we have for any T > 0. To prove the theorem we show that any nontrivial solution of (3.9)-(3.11) blows up in finite time. We set Integrating (3.9) over Ω and using Green's identity and Jensen's inequality, we have Integrating last inequality, we obtain the desired result due to (3.6). Proof. Let w(x, t) be a solution of the following problem   (3.12) By a direct computation we can check that is a supersolution of (1.1)-(1.3) in Q T for any T > 0. To prove the theorem we show the existence of global bounded solutions of (3.12). Let us consider the following auxiliary linear problem (3.13) As it was proved in [8] any solution of (3.13) is a bounded function. Now we construct a supersolution of (3.12) in the following form g(x, t) = ah(x, t), where a is some positive constant. It is obvious, for small values of a. Then by a comparison principle for (3.12) and suppose that where t 0 is some positive constant, Let us define f (t) as a solution of the following equation Then f (t) can be written in an explicit form (3.20) We rewrite (3.20) as following (3.21) We prove that right hand side I of (3.21) is bounded below by some positive constant. The numerator and the denominator of I tend to infinity as t → ∞ by virtue of (3.15). Using (3.18) we can obtain that lim inf By (3.20) - (3.22) we conclude that ≤ Ω k(x, y, t)u l (y, t) dy, x ∈ ∂Ω, t > t 1 (3.27) for large values of t 1 . By (3.17), (3.23) -(3.27) and Theorem 2.3 for some d 2 > 0 and any T > t 1 if We set Integrating (1.1) over Ω and using (3.14), (3.16), (3.28), (3.29) and Green's identity, we have where t ≥ t 2 , t 2 is large enough and lim t→∞ ξ(t) dt = ∞. Integrating (3.30) over (t 2 , t) we find that Now we deduce lower bound for k(t). From (3.18) we conclude that for some t 3 ≥ t 2 . By (3.16), (3.32) we have where lim t→∞ γ i (t) = ∞, i = 1, 2. Let us change unknown function It is easy to check that w(x, t) is a solution of the following problem ∂w(x, t) ∂ν = exp (l − 1)t m Ω k(x, y, t)w l (y, t) dy, x ∈ ∂Ω, t ≥ 0, (3.36) u(x, 0) = u 0 (x), x ∈ Ω.
To prove the optimality of (3.16) for blow-up of any nontrivial solution of (1.1)- and assume that  Proof. To prove the theorem we construct a supersolution of (1.1)-(1.3) in Q T for any T > 0. Let us define g(t) as a positive solution of the following equation where b was defined in (3.3). Then g(t) can be written in an explicit form .
Hence, u(x, t) cannot blow up in Ω ′ × [0, T ]. Since Ω ′ is an arbitrary subset of Ω, the proof is completed.
From [27] it is easy to get the following result.