Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up time of the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$ and the lower blow-up estimates of the solutions.

We set L ∞ uloc,ρ (Ω) = L ∞ (Ω) and L ∞ uloc,ρ (Ω) = BU C(Ω). In this paper we prove the local existence and the uniqueness of the solutions of problem (1.1) with initial functions in L r uloc,ρ (Ω), and study the dependence of the blow-up time on the initial functions. As an application of the main results of this paper, we study the asymptotic behavior of the blow-up time T (ϕ) with ϕ = λψ as λ → 0 or λ → ∞ and show the validity of our arguments. Furthermore, we obtain a lower estimate of the blow-up rate of the solutions (see Section 5).
Next we state the definition of the solution of (1.1).
These follow from property (i) in Section 2.
Now we are ready to state the main results of this paper. Let p * = 1 + 1/N .
there exists a positive constant γ 1 such that, for any ϕ ∈ L r uloc,ρ (Ω) with (1.10) Here C and µ are constants depending only on N , Ω, p and r.
Theorem 1.2 Assume the same conditions as in Theorem 1.1. Let v and w be L r uloc (Ω)- (1.11) Then there exists a positive constant γ 2 such that, if We give some comments related to Theorems 1.1 and 1.2.
(ii) Consider the case Ω = R N + . Let u be a L r uloc (Ω)-solution of (1.1) blowing up at t = T < ∞, where r is as in (1.7). Then, for any µ > 0, u µ defined by (1.2) satisfies and it blows up at t = µ −2 T . This means that Theorem 1.1 holds with ρ = 1 if and only if Theorem 1.1 holds for any ρ > 0.
then, for any γ > 0, we can find a constant ρ > 0 such that ρ As a corollary of Theorem 1.1, we have: Assume the same conditions as in Theorem 1.1 and p > p * .
(ii) Assume ρ * = ∞. Then there exists a constant γ such that, if For further applications of our theorems, see Section 5. [7] and [14]. For the case p > p * , it is proved in [24] (see also [23]) that, if ϕ ≥ 0, ϕ ≡ 0 in Ω and is sufficiently small, then there exists a positive global-in-time solution of (1.1). This also immediately follows from assertion (ii) of Corollary 1.1 and the comparison principle.
We explain the idea of the proof of Theorem 1.1. Under the assumptions of Theorem 1.1, there exists a sequence {ϕ n } ∞ n=1 ⊂ BU C(Ω) such that lim n→∞ ϕ − ϕ n r,ρ = 0, sup n ϕ n r,ρ ≤ 2 ϕ r,ρ . (1.14) For any n = 1, 2, . . . , let u n satisfy in the classical sense in Ω, (1.15) where T n is the blow-up time of the solution u n . By regularity theorems for parabolic equations (see e.g. [8] and [25, Chapters III and IV]) we see that for any 0 < τ < T < T n , which imply that u n is a L r uloc (Ω)-solution in Ω × [0, T n ) for any 1 ≤ r < ∞. Set |u n (y, τ )| r dy, 0 ≤ t < T n .
It follows from (1.8) and (1.14) that where M is the integer given in Lemma 2.1. We adapt the arguments in [2], [3] and [22] to obtain uniform estimates of u n and u m − u n with respect to m, n = 1, 2, . . . , and prove that inf n T * n ≥ µρ 2 , inf n T * * n ≥ µρ 2 , for some µ > 0. This enables us to prove Theorem 1.1. Theorem 1.2 follows from a similar argument as in Theorem 1.1.
The rest of this paper is organized as follows. In Section 2 we give some preliminary lemmas related to ρ * . In Sections 3 and 4 we prove Theorems 1.1 and 1.2. In Section 5, as applications of Theorem 1.1, we give some results on the blow-up time and the blow-up rate of the solutions.

Preliminaries
In this section we recall some properties of uniformly local L r spaces and prove some lemmas related to ρ * . Furthermore, we give some inequalities used in Sections 3 and 4. In what follows, the letter C denotes a generic constant independent of x ∈ Ω, n and ρ.
Let 1 ≤ r < ∞. We first recall the following properties of L r uloc,ρ (Ω): (i) if f ∈ L r uloc,ρ (Ω) for some ρ > 0, then, for any ρ ′ > 0, f ∈ L r uloc,ρ ′ (Ω) and f r,ρ ′ ≤ C 1 f r,ρ for some constant C 1 depending only on N , ρ and ρ ′ ; (ii) there exists a constant C 2 depending only on N such that for any 1 ≤ r ≤ q < ∞ and ρ > 0; (iii) if f ∈ L r (Ω), then f ∈ L r uloc,ρ (Ω) for any ρ > 0 and lim Properties (ii) and (iii) are proved by the Hölder inequality and the absolute continuity of |f | r dy with respect to dy. Property (i) follows from the following lemma.
Let v and w be L r uloc (Ω)-solutions of (1.1) in Ω × [0, T ], where 0 < T < ∞ and r is as in (1.7). Set z := v − w and z ǫ := max{z, 0} + ǫ for ǫ ≥ 0. Then z ǫ satisfies in the weak sense (see e.g. [9, Chapter II]). Here In this section we give some estimates of z, and prove Theorems 1.1 and 1.2 in the case r > 1.
On the other hand, for the case 1 ≤ r < 2, applying (3.13) with r = 2 to the cylinders Q j and Q j+1 , we have where b = 2 (N +2)/2 . Then, for any ν > 0, we have Taking a sufficiently small ν if necessary, we see that for some ρ ∈ (0, ρ * /2), then for all 0 ≤ τ ≤ t ≤ T with t − τ ≤ µρ 2 , where C and µ are positive constants depending only on N , Ω, p and r.
Proof. Let x ∈ Ω and ζ be as in Lemma 2.4. Let k be as in Lemma 2.4 and ǫ > 0. Similarly to (3.18), we have for all 0 < τ < t ≤ T with t − τ ≤ µρ 2 . Then we deduce from (3.29)-(3.31) that Then, taking sufficiently small constants Λ and µ if necessary, we obtain for all 0 < τ < t ≤ T with t − τ ≤ µρ 2 . This implies (3.26), and the proof is complete. ✷ Now we are ready to complete the proof of Theorems 1.1 and 1.2 in the case r > 1.

Applications
In this section, as an application of Theorem 1.1, we give lower estimates of the blow-up time and the blow-up rate for problem (1.1).

Blow-up time
Let T (λψ) be the blow-up time of the solution of (1.1) with the initial function ϕ = λψ.
In this subsection we study the behavior of T (λψ) as λ → ∞ or λ → 0.
Theorem 5.1 Let N ≥ 1 and Ω ⊂ R N be a uniformly regular domain of class C 1 . Let r satisfy Then, for any ψ ∈ L r uloc,ρ (Ω) with ρ > 0, there exists a positive constant C such that for all sufficiently large λ.
Proof. Let γ 1 and µ be constants given in Theorem 1.1. If r < ∞, by Theorem 1.1 we see that for all sufficiently large λ. If r = ∞, then It follows from Theorem 1.1 that for all sufficiently large λ. Thus Theorem 5.1 follows. ✷ Theorem 5.2 Let N ≥ 1 and Ω ⊂ R N be a uniformly regular domain of class C 1 . Assume Then there exists a positive constant C 1 such that for some δ > 0, then there exists a positive constant C 2 such that for all sufficiently large λ.
Proof. In the case 1 < p < p * , let r > 1, r > N (p − 1) and β < N/r. In the case 1 < p < p * , let r = 1. It follows from (5.1) that ρ where A is a positive constant to be chosen as ψ(x) ≥ v(x, 0) in R N + . By [7, Lemma 2.1.2] we can find a constant c p depending only on p such that On the other hand, since T (λψ) ≤ T (λv(0)) and Motivated by [26], we consider the case Ω = R N + and study the behavior of the blow-up time T (λψ) as λ → 0. for all sufficiently small λ > 0, where then there exists a positive constant C 2 such that T (λψ) ≤ C 2 f (λ) (5.8) for all sufficiently small λ > 0.
Let v be a solution of where A is a positive constant to be chosen as ψ(x) ≥ v(x, 0) in R N + . Since T (λψ) ≤ T (λv(0)) and v(·, 0, t) L ∞ (R N−1 ) ≥ for all sufficiently large t, by a similar argument as in the proof of (5.4) we obtain (5.8). Thus Theorem 5.3 follows. ✷

Blow-up rate
Let u be a solution of (1.1) in Ω × [0, T ), where 0 < T < ∞, such that u blows up at t = T . In this subsection, as a corollary of Theorem 1.1, we state a result on lower estimates of the blow-up rate of the solution u. Blow-up rate of positive solutions for problem (1.1) was first obtained by Fila and Quittner [12], where it was shown that lim sup t→T (T − t) 1 2(p−1) u(t) L ∞ (Ω) < ∞ (5.9) holds in the case where Ω is a ball, the initial function ϕ is radially symmetric and satisfies some monotonicity assumptions. Subsequently, it was proved that (5.9) holds for positive solutions in the following cases: • Ω is a bounded smooth domain, (N − 2)p < N and ∂ t u ≥ 0 in Ω × (0, T ) (see [16], [18] and [21]); • Ω is a bounded smooth domain and p ≤ 1 + 1/N (see [20]); • Ω = R N + and (N − 2)p < N (see [5]). See [30] for sign changing solutions. On the other hand, for positive solutions, it was shown in [21]  holds if Ω is a bounded smooth domain (see also [16] and [18]).
We state a result on lower estimates of the blow-up rate of the solutions. Theorem 5.4 is a generalization of (5.10) and it holds without the boundedness of the domain Ω and the positivity of the solutions. This implies (5.11) in the case r < ∞. Furthermore, by (5.13), for any t ∈ (T − δ, T ), there exist x(t) ∈ Ω and y(t) ∈ Ω(x(t), ρ(t)) such that Cρ(t) N u(y(t), t) r ≥ Ω(x(t),ρ(t)) u(y, t) r dy ≥ This yields (5.11) in the case r = ∞, and Theorem 5.4 follows. ✷