Global well-posedness of solutions to a class of heat equation under dynamical type boundary conditions

Mahmoud El Ahmadi can be contacted at: mahmoud.elahmadi@usmba.ac.ma

Conflict of interest: All authors declare that they have no conflicts of interest.

Purpose

By employing the Galerkin approximation and a family of potential wells, we establish the existence of a global solution and finite-time blow-up under certain suitable conditions. Additionally, we provide results concerning the asymptotic behavior of certain solution with positive initial energy.

Design/methodology/approach

The authors used the Galerkin approximation method and potential-well method.

Findings

The authors established some results on the existence of a global solution, the finite time blow-up of solutions and the asymptotic behavior of certain solutions with positive initial energy.

Originality/value

This work generalizes several published works.

1. Introduction

In this work, we mainly study the following parabolic problem:

\{\begin{cases} \frac{\partial u}{\partial t} - Δ u + u = 0 in Ω, t > 0, \\ σ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial ν} = f (u) on \partial Ω, t > 0, \\ u (x, 0) = u_{0} (x) in Ω, \end{cases}

(1)

where $Ω \subset R^{N} (N \geq 2)$ is a bounded domain with smooth boundary ∂Ω, σ > 0 and f(u) satisfies the conditions as follows:

(C) \{\begin{cases} f \in C^{1} and f (0) = f^{'} (0) = 0, \\ f (u) is monotone, and is convex for u > 0, concave for u < 0, \\ (p + 1) F (u) \leq u f (u), | u f (u) | \leq β | F (u) |, \end{cases}

(2)

where

F (u) = \int_{0}^{u} f (s) d s,

and

\{\begin{cases} 2 < p + 1 \leq β < \infty if N = 2, \\ 2 < p + 1 \leq β \leq \frac{2 (N - 1)}{N - 2} if N \geq 3 . \end{cases}

Parabolic problems have many applications in various fields of mathematics. Over the last decade, they have become a popular subject of research in partial differential equations. These types of problems appear quite naturally, notably in the fields of diffusion of heat in materials, the propagation of acoustic or electromagnetic waves and thermal conduction. In addition, parabolic problems can be used to model the growth and propagation of a population in a given environment. For example, these problems can be used to study the spread of an infectious disease in a population, the diffusion of nutrients in a biological tissue and the migration of species. (see Refs. [1–8]) and references therein. The interesting aspect of this paper is the dynamical boundary condition, which connects the outer normal derivative with the time derivative:

σ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial ν} = f (u) .

In general, the value of dynamic boundary conditions lies in their ability to capture realistic, dynamic phenomena that cannot be adequately represented by static conditions. They are particularly useful for modeling complex and varied physical processes, where boundary conditions may change in response to external events or dynamic interactions with the environment (see Refs. [9–11]).

Many papers have dealt with the heat equations (see Refs. [12–18]). For example, Wang et al., in Ref. [18], studied the Cauchy problem:

\{\begin{cases} u_{t} - Δ u - u^{α} + u = 0, x \in R^{N}, t > 0, \\ u (x, 0) = Ψ (x) \geq 0, \end{cases}

where 1 < α < ∞ if N = 1, 2 and $1 < α \leq \frac{N + 2}{N - 2}$ if N ≥ 3. Letting v(x) be the steady state solution to the above problem, they proved that:

If Ψ(x) ≤ v(x), then problem has a global L^p solution.
If Ψ(x) ≤ v(x) and Ψ(x) ≠ v(x), then the L^p solution u(t) → 0 as t → +∞.
If Ψ(x) ≥ v(x) and Ψ(x) ≠ v(x), then the L^p solution blows up in finite time.

Also, L. E. Payne and P. W. Schaefer, in Ref. [17], considered the heat equation subject to a nonlinear boundary condition, i.e.

\{\begin{cases} \frac{\partial u}{\partial t} - Δ u = 0 in Ω, t > 0, \\ \frac{\partial u}{\partial ν} = f (u) on \partial Ω, t > 0, \\ u (x, 0) = g (x) \geq 0 in Ω, \end{cases}

where Ω is a bounded smooth convex domain in $R^{3}$ and f satisfies the condition

0 \leq f (s) \leq k s^{(n + 2) / 2}, s > 0,

for some positive constants k and n ≥ 1. By using a differential inequality technique, the authors determined a lower bound on the blow-up time for solutions of the heat equation when the solution explosion occurs. In addition, a sufficient condition which implies that blow-up does occur is determined.

Recently, in Ref. [15], A. Lamaizi et al. have considered the following problem:

\{\begin{cases} u_{t} - Δ u + u = 0 in Ω \times (0, T), \\ \frac{\partial u}{\partial ν} = λ | u |^{p - 1} u on \partial Ω \times (0, T), \\ u (x; 0) = u_{0} (x) in Ω, \end{cases}

where $Ω \subset R^{n}$ is an open bounded domain for n ≥ 2 with smooth boundary ∂Ω, λ > 0 and p satisfies

(H) \{\begin{cases} 1 \leq p \leq \frac{n}{n - 2} if n > 2, \\ 1 \leq p < \infty if n = 2 . \end{cases}

By using the Galerkin approximation, they established the existence of global weak solution and finite time blow-up under some suitable conditions. So, a natural question arises, can we obtain some qualitative results such as the existence and blow up of solutions if we replace the non-linear condition imposed on the boundary by the following dynamic condition $σ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial ν} = f (u)$ where σ > 0 and the function f(u) satisfies condition (C)? Then, the goal of this article is to give a positive answer to this question, more precisely, we will establish the existence and blow up results by applying Galerkin approximation and similar techniques to those used in Ref. [15].

2. Preliminaries

Throughout this work, we designate the Lebesgue space L^p(Ω) by:

L^{p} (Ω) = \{u : Ω \to R measurable such that \int_{Ω} | u (x) |^{p} d x < + \infty\}

equipped with the norm

‖ u ‖_{p} = {(\int_{Ω} | u (x) |^{p} d x)}^{\frac{1}{p}} .

For p = ∞, we denote

L^{\infty} (Ω) = \{u : Ω \to R measurable such that ess - \sup_{Ω} | u | < + \infty\}

with

ess - \sup_{Ω} | u | = \inf {C > 0 such that | u (x) | \leq C a.e. Ω},

and

L^{p} (\partial Ω, ρ) = \{u : \partial Ω \to R measurable such that {(\int_{\partial Ω} | u (x) |^{p} ρ (x) d σ (x))}^{1 / p} < + \infty\}

where: ρ: ∂Ω → [0, ∞) is a given weight function, and dσ is the surface measure (or the Hausdorff measure) on ∂Ω.

For p = ∞, the space is defined as:

L^{\infty} (\partial Ω, ρ) = \{u : \partial Ω \to R ∣ ess \sup_{x \in \partial Ω} | u (x) | < \infty\}

with the norm:

‖ u ‖_{L^{\infty} (\partial Ω, ρ)} = ess \sup_{x \in \partial Ω} | u (x) | ρ (x)

Especially, for p = 2, the scalar product of L²(Ω) will be denoted by 〈⋅, ⋅〉 and the scalar product of L²(∂Ω, ρ) will be denoted by 〈⋅,⋅〉₀:

〈 u, v 〉 = \int_{Ω} u v d x, {〈 u, v 〉}_{0} = \oint_{\partial Ω} u v d ρ .

Moreover, usual Sobolev space on Ω is defined by

W^{1,2} (Ω) = \{u \in L^{2} (Ω) : | ▽ u | \in L^{2} (Ω)\},

and it is equipped with the norm

‖ u ‖_{1,2}^{2} = ‖ u ‖_{2}^{2} + ‖ ▽ u ‖_{2}^{2} .

Recall the following embedding result.

Lemma 1.

(See Ref. [10]) The trace operator u: W^1,q(Ω) → L^r(∂Ω, ρ) is continuous if and only if

\{\begin{cases} 1 \leq r \leq q^{\partial} if q \neq N, \\ 1 \leq r < \infty if q = N, \end{cases}

where

q^{\partial} ≔ \{\begin{cases} \frac{q (N - 1)}{N - q} if 1 < q < N, \\ \infty if q \geq N . \end{cases}

Define the space

X^{p} = L^{p} (Ω) \times L^{p} (\partial Ω, ρ), for 1 \leq p \leq \infty,

endowed with the norm

‖ U ‖_{X^{p}} ≔ {(‖ u ‖_{p}^{p} + σ ‖ φ ‖_{p, \partial Ω}^{p})}^{1 / p},

for $U = (u, φ) \in X^{p}$ and σ > 0.

In particular for p = 2, we denote

{〈 \frac{\partial u}{\partial t}, φ 〉}_{σ} ≔ 〈 \frac{\partial u}{\partial t}, φ 〉 + σ {〈 \frac{\partial u}{\partial t}, φ 〉}_{0},

for any φ ∈ W^1,p(Ω), and

{‖\frac{\partial u}{\partial t}‖}_{σ}^{2} ≔ {‖\frac{\partial u}{\partial t}‖}_{2}^{2} + σ {‖\frac{\partial u}{\partial t}‖}_{2, \partial Ω}^{2} .

Let X be a Banach space and T > 0. Denote the following spaces:

C ([0, T]; X) = {u : [0, T] \to X continue},

L^{p} (0, T; X) = \{u : [0, T] \to X is a measurable such that \int_{0}^{T} ‖ u (t) ‖_{X}^{p} d t < \infty\},

equipped with the norm

‖ u ‖_{L^{p} (0, T; X)} = {(\int_{0}^{T} ‖ u (t) ‖_{X}^{p} d t)}^{\frac{1}{p}},

and

L^{\infty} (0, T; X) = \{u : [0, T] \to X is a measurable such that : \exists C > 0; ‖ u (t) ‖_{X} < C a.e.t\},

endowed with the norm

‖ u ‖_{L^{\infty} (0, T; X)} = \inf \{C > 0; ‖ u (t) ‖_{X} < C a.e.t\} .

We define some functionals and sets as follows

I (u) = \frac{1}{2} ‖ u ‖_{1,2}^{2} - \int_{\partial Ω} F (u) d ρ,

J (u) = ‖ u ‖_{1,2}^{2} - \int_{\partial Ω} u f (u) d ρ,

S = \{u \in W^{1,2} (Ω) ∣ J (u) > 0, I (u) < h\} \cup {0},

U = \{u \in W^{1,2} (Ω) ∣ J (u) < 0, I (u) < h\},

where

h = \inf_{u \in Y} I (u),

and

Y = \{u \in W^{1,2} (Ω) ∣ J (u) = 0, ‖ u ‖_{1,2} \neq 0\} .

Next, we shall introduce a family of potential wells S_λ for λ > 0, and its corresponding sets U_λ and we always assume that f satisfies (C).

S_{λ} = \{u \in W^{1,2} (Ω) ∣ J_{λ} (u) > 0, I (u) < h (λ)\} \cup {0},

U_{λ} = \{u \in W^{1,2} (Ω) ∣ J_{λ} (u) < 0, I (u) < h (λ)\},

where

J_{λ} (u) = λ ‖ u ‖_{1,2}^{2} - \int_{\partial Ω} u f (u) d ρ,

h (λ) = \inf_{u \in Y_{λ}} I (u),

and

Y_{λ} = \{u \in W^{1,2} (Ω) ∣ J_{λ} (u) = 0, ‖ u ‖_{1,2} \neq 0\},

In the present paper, our purpose is to study the global well-posedness, asymptotic behavior and finite time blow-up of solutions for problem (1). The Nehari functional, born from the studies of the elliptic equations in the nonlinear analysis [19, 20] plays a very important role to divide the manifold of the initial data leading to global existence and finite time blow-up. In this paper we define a family of potential wells by using the energy functional I(u) and the auxiliary functional J(u) so that it is easy to show the structure of the potential wells and the sufficient conditions for global well-posedness, asymptotic behavior and finite time blow-up of solutions for problem (1).

3. Global existence of solutions

This section is devoted to showing the existence of solutions to our problem. Before giving the first result, we give the definition of weak solution and state some lemmas which will be used later.

Definition 1.

Let T > 0. A function $u = u (x, t) \in L^{\infty} (0, \infty; W^{1,2} (Ω)) \cap C ([0, T]; L^{2} (Ω) \times L^{2} (\partial Ω, ρ))$ with $\frac{\partial u}{\partial t} \in L^{2} (0, \infty; L^{2} (Ω))$ is said to be a weak solution to the problem (1) in Ω × [0, T), if u(x, 0) = u₀ ∈ W^1,2(Ω), and satisfies

{〈 \frac{\partial u}{\partial t}, v 〉}_{σ} + 〈 u, v 〉 + 〈 ▽ u, ▽ v 〉 = {〈 f (u), v 〉}_{0}, \forall v \in W^{1,2} (Ω), t \in (0, T) .

Moreover,

\int_{0}^{t} {‖\frac{\partial u}{\partial t}‖}_{σ}^{2} d τ + I (u) \leq I (u_{0}), \forall t \in [0, T) .

(3)

Lemma 2.

([7 ]). Suppose that (C) holds. Then

|F(u)|≤M|u|^β for some M > 0 and all $u \in R$ ⁠.
F(u) ≥ N|u|^p+1 for some N > 0 and |u|≥1.
The equality $u (u f^{'} (u) - f (u)) \geq 0$ holds only for u = 0.

As a result, the following corollary is obtained.

Corollary 1.

Let f(u) satisfy (C). Then

|uf(u)|≤βM|u|^β, |f(u)|≤βM|u|^β−1 for all $u \in R$ ⁠.
uf(u)≥(p + 1)N|u|^p+1 for |u|≥1.

Lemma 3.

Suppose that 0 < I(u) < h for some u ∈ W^1,2(Ω), λ₁ < λ₂ are the two roots of equation h(λ) = I(u). Then the sign of J_λ(u) does not change for λ₁ < λ < λ₂.

Proof. Arguing by contradiction, we assume that the sign of J_λ(u) is changeable for λ₁ < λ < λ₂, then there exist a $λ_{0} \in (λ_{1}, λ_{2})$ such that $J_{λ_{0}} (u) = 0$ ⁠. From I(u) > 0 we get ‖u‖_1,2 ≠ 0, hence $u \in Y_{λ_{0}}$ ⁠, consequently I(u) ≥ h(λ₀), which contradicts

I (u) = h (λ_{1}) = h (λ_{2}) < h (λ_{0}) .

□

Lemma 4.

Suppose that (C) holds, u₀(x) ∈ W^1,2(Ω), 0 < e < h and λ₁ < λ₂ be the two roots of equation h(λ) = e. If $J (u_{0}) > 0$ , then all weak solutions u(t) of problem (1) with $I (u_{0}) = e$ belong to S_λ for λ₁ < λ < λ₂ and $t \in [0, T)$ ⁠.

Proof. By $I (u_{0}) = e, J (u_{0}) > 0$ and Lemma 3, we can deduce $J_{λ} (u_{0}) > 0$ and $I (u_{0}) < h (λ)$ i.e. u₀(x) ∈ S_λ for λ₁ < λ < λ₂. Let u(t) be any weak solution of problem (1) with $I (u_{0}) = e$ and $J (u_{0}) > 0$ ⁠, and T be the maximal existence time of u(t). Arguing by contradiction, we suppose that there exist a $λ_{0} \in (λ_{1}, λ_{2})$ and t₀ ∈ (0, T) such as $J_{λ_{0}} (u (t_{0})) = 0, {‖u (t_{0})‖}_{1,2} \neq 0$ or $I (u (t_{0})) = h (λ_{0})$ ⁠. From (3), we conclude that

\int_{0}^{t} {‖\frac{\partial u}{\partial τ}‖}_{σ}^{2} d τ + I (u) \leq I (u_{0}) < h (λ), λ_{1} < λ < λ_{2}, 0 \leq t < T .

(4)

Therefore $I (u (t_{0})) \neq h (λ_{0})$ ⁠. If $J_{λ_{0}} (u (t_{0})) = 0$ and ${‖u (t_{0})‖}_{1,2} \neq 0$ ⁠, thus the definition of h(λ) implies that $I (u (t_{0})) \geq h (λ_{0})$ ⁠, which contradicts (4). □

Theorem 1.

Let u₀(x) ∈ W^1,2(Ω) and f(u) satisfy (C). Suppose that 0 < I(u₀) < h and $J (u_{0}) > 0$ ⁠. Then, problem (1) admits a global weak solution $u (t) \in L^{\infty} (0, \infty; W^{1,2} (Ω)) \cap C ([0, T]; L^{2} (Ω) \times L^{2} (\partial Ω, ρ))$ with $\frac{\partial u}{\partial t} \in L^{2} (0, \infty; L^{2} (Ω))$ and u(t) ∈ S for 0 ≤ t < ∞.

Proof of Theorem 1: We shall use the Galerkin method of approximation (for more information see Refs. [21, 22]). Let w_j(x) be a system of base functions in W^1,2(Ω). Define the approximate solutions u_m(x, t) of problem (1):

u_{m} (x, t) = \sum_{j = 1}^{m} Φ_{j m} (t) w_{j} (x), m = 1,2, \dots

verifying

{〈 \frac{\partial u_{m}}{\partial t}, w_{s} 〉}_{σ} + 〈 u_{m}, w_{s} 〉 + 〈 ▽ u_{m}, ▽ w_{s} 〉 = {〈 f (u_{m}), w_{s} 〉}_{0}, s = 1,2, \dots, m

(5)

u_{m} (x, 0) = \sum_{j = 1}^{m} a_{j m} w_{j} (x) \to u_{0} (x) in W^{1,2} (Ω) .

(6)

Multiplying (5) by $Φ_{s m}^{'} (t)$ and summing for s yields

\int_{0}^{t} {‖\frac{\partial u_{m}}{\partial t}‖}_{σ}^{2} d τ + I (u_{m}) \leq I (u_{0}) < h, \forall t \in [0, T),

(7)

and u_m ∈ S for sufficiently large m and t ≥ 0 (see the proof of Lemma 4).

Combining (7) and

I (u_{m}) = \frac{1}{2} ‖ u ‖_{1,2}^{2} - \int_{\partial Ω} F (u_{m}) d ρ \geq \frac{1}{2} ‖ u ‖_{1,2}^{2} - \frac{1}{p + 1} \int_{\partial Ω} u_{m} f (u_{m}) d ρ = (\frac{1}{2} - \frac{1}{p + 1}) ‖ u ‖_{1,2}^{2} + \frac{1}{p + 1} J (u_{m}) \geq \frac{p - 1}{2 (p + 1)} ‖ u ‖_{1,2}^{2},

we obtain

\int_{0}^{t} {‖\frac{\partial u_{m}}{\partial t}‖}_{σ}^{2} d τ + \frac{p - 1}{2 (p + 1)} ‖ u_{m} ‖_{1,2}^{2} < h, 0 \leq t < \infty

(8)

for sufficiently large m.

Using (8), we have

‖ u_{m} ‖_{1,2}^{2} < \frac{2 (p + 1)}{p - 1} h, 0 \leq t < \infty,

(9)

{‖u_{m}‖}_{β, \partial Ω}^{2} \leq C_{*}^{2} ‖ u_{m} ‖_{1,2}^{2} < C_{*}^{2} \frac{2 (p + 1)}{p - 1} h, 0 \leq t < \infty,

(10)

where C_* is the embedding constant form W^1,2(Ω) into L^β(∂Ω, ρ).

\begin{aligned} {‖f (u_{m})‖}_{q, \partial Ω}^{q} & \leq \int_{\partial Ω} {(β M {|u_{m}|}^{β - 1})}^{q} d ρ \\ = {(β M)}^{q} {‖u_{m}‖}_{β, \partial Ω}^{β} \\ \leq {(β M)}^{q} C_{*}^{β} {(\frac{2 (p + 1)}{p - 1} h)}^{β / 2}, q = \frac{β}{β - 1}, 0 \leq t < \infty, \end{aligned}

(11)

and

\int_{0}^{t} {‖\frac{\partial u_{m}}{\partial t}‖}_{σ}^{2} d τ < h, 0 \leq t < \infty .

(12)

Therefore, there exist u, ϕ and a subsequence $\{u_{v}\}$ of $\{u_{m}\}$ such that,

u_v → u in $L^{\infty} (0, \infty; W^{1,2} (Ω))$ weakly star and a.e. in Ω × [0, ∞),

u_vt → u_t in $L^{2} (0, \infty; L^{2} (Ω))$ weakly star,

$f (u_{v}) \to ϕ$ in $L^{\infty} (0, \infty; L^{q} (\partial Ω))$ weakly star and a.e. in ∂Ω × [0, ∞).

Consequently, from Lemma 1.3 in Ref. [23], we obtain ϕ = f(u). In (5) for fixed s letting m = v → ∞ we have

{〈 \frac{\partial u}{\partial t}, w_{s} 〉}_{σ} + 〈 u, w_{s} 〉 + 〈 ▽ u, ▽ w_{s} 〉 = {〈 f (u), w_{s} 〉}_{0} \forall s,

and

{〈 \frac{\partial u}{\partial t}, v 〉}_{σ} + 〈 u, v 〉 + 〈 ▽ u, ▽ v 〉 = {〈 f (u), v 〉}_{0} \forall v \in W^{1,2} (Ω), t > 0 .

By (6), we obtain u(x, 0) = u₀(x) in W^1,2(Ω). Then u(x, t) is a global weak solution of problem (1). Finally, from Lemma 4, we have u(t) ∈ S, for t ≥ 0. The proof of the Theorem 1 is now finished.

4. Blow up in finite time

In this part, we prove the blow-up of solutions to problem (1). Firstly, we give the following lemmas.

Lemma 5.

Let f(u) satisfy (C). Assume that J_λ(u) < 0, then ‖u‖_1,2 > z(λ). In particular, if J(u) < 0, then ‖u‖_1,2 > z(1).

Where

z (λ) = {(\frac{λ}{a C_{*}^{β}})}^{1 / (β - 2)},

and

a = \sup \frac{u f (u)}{| u |^{β}} .

Proof. J_λ(u) < 0 gives

λ ‖ u ‖_{1,2}^{2} < \int_{\partial Ω} u f (u) d ρ \leq a ‖ u ‖_{β, \partial Ω}^{β} \leq a C_{*}^{β} ‖ u ‖_{1,2}^{β - 2} ‖ u ‖_{1,2}^{2} .

(13)

Consequently, (13) implies ‖u‖_1,2 > z(λ). □

Lemma 6.

Suppose that (C) holds,u₀(x) ∈ W^1,2(Ω) and 0 < e < h, where λ₁ < λ₂ are the two roots of equation h(λ) = e. Suppose that $J (u_{0}) < 0$ , then all weak solutions of problem (1) with $I (u_{0}) = e$ belong to U_λ for λ ∈ (λ₁, λ₂).

Proof. Let u(t) be any solution of problem (1) with $I (u_{0}) = e$ and $J (u_{0}) < 0, T$ be the existence time of u(t). First from $I (u_{0}) = e$ ⁠, J(u₀) < 0 and Lemma 3 we conclude that $J_{λ} (u_{0}) < 0$ and $I (u_{0}) < h (λ)$ ⁠, i.e. u₀(x) ∈ U_λ for λ₁ < λ < λ₂. Next we prove u(t) ∈ U_λ for λ₁ < λ < λ₂ and 0 < t < T. If it is false, let t₀ ∈ (0, T) be the first time such that u(t) ∈ U_λ for 0 ≤ t < t₀ and $u (t_{0}) \in \partial U_{λ}$ ⁠, i.e. $J_{λ} (u (t_{0})) = 0$ or $I (u (t_{0})) = h (λ)$ for some $λ \in (λ_{1}, λ_{2})$ ⁠. So (4) implies $I (u (t_{0})) = h (λ)$ is impossible. If $J_{λ} (u (t_{0})) = 0$ ⁠, thus J_λ(u(t)) < 0 for 0 < t < t₀ and Lemma 5 yield ‖u(t)‖_1,2 > z(λ) and ${‖u (t_{0})‖}_{1,2} \geq z (λ)$ ⁠. It follows from the definition of h(λ) that $I (u (t_{0})) \geq h (λ)$ which contradicts (4). □

Theorem 2.

Let u₀(x) ∈ W^1,2(Ω) and f(u) satisfy (C). Assume that $I (u_{0}) < h$ and $J (u_{0}) < 0$ ⁠. Then, the solution of problem (1) must blow up in finite time i.e. there exists a T > 0 such that

\lim_{t \to T} \int_{0}^{t} ‖ u ‖_{σ}^{2} d τ = + \infty .

(14)

Proof of Theorem 2: Let u(t) be any solution of problem (1) with $I (u_{0}) < h$ and $J (u_{0}) < 0$ ⁠. We consider the auxiliary function

φ_{1} (t) = \int_{0}^{t} ‖ u ‖_{σ}^{2} d τ .

A direct calculation gives

{\dot{φ}}_{1} (t) = ‖ u ‖_{σ}^{2},

and

{\ddot{φ}}_{1} (t) = 2 {〈 \frac{\partial u}{\partial t}, u 〉}_{σ} = 2 (〈 f (u), u 〉 - ‖ u ‖_{1,2}^{2}) = - 2 J (u) .

(15)

By (15), (3) and

\int_{\partial Ω} u f (u) d ρ \geq (p + 1) \int_{\partial Ω} F (u) d ρ

we have

{\ddot{φ}}_{1} (t) \geq 2 (p + 1) \int_{0}^{t} {‖\frac{\partial u}{\partial t}‖}_{σ}^{2} d τ + (p - 1) ‖ u ‖_{1,2}^{2} - 2 (p + 1) I (u_{0}) \geq 2 (p + 1) \int_{0}^{t} {‖\frac{\partial u}{\partial t}‖}_{σ}^{2} d τ + (p - 1) {\dot{φ}}_{1} (t) - 2 (p + 1) I (u_{0}),

and

φ_{1} {\ddot{φ}}_{1} - \frac{p + 1}{2} {({\dot{φ}}_{1})}^{2} ⩾ 2 (p + 1) [\int_{0}^{t} ‖ u ‖_{σ}^{2} d τ \int_{0}^{t} {‖\frac{\partial u}{\partial t}‖}_{σ}^{2} d τ - {(\int_{0}^{t} {〈 u, \frac{\partial u}{\partial t} 〉}_{σ} d τ)}^{2}] + (p - 1) φ_{1} {\dot{φ}}_{1} - (p + 1) {‖u_{0}‖}_{σ}^{2} {\dot{φ}}_{1} - 2 (p + 1) I (u_{0}) φ_{1} + \frac{p + 1}{2} {‖u_{0}‖}_{σ}^{2} .

According to Hölder inequality, we deduce that

\begin{aligned} φ_{1} {\ddot{φ}}_{1} - \frac{p + 1}{2} {({\dot{φ}}_{1})}^{2} \geq & (p - 1) φ_{1} {\dot{φ}}_{1} - (p + 1) {‖u_{0}‖}_{σ}^{2} {\dot{φ}}_{1} \\ - 2 (p + 1) I (u_{0}) φ_{1} + \frac{p + 1}{2} {‖u_{0}‖}_{σ}^{2} . \end{aligned}

(16)

If $I (u_{0}) \leq 0$ ⁠, then

φ_{1} {\ddot{φ}}_{1} - \frac{p + 1}{2} {({\dot{φ}}_{1})}^{2} \geq (p - 1) φ_{1} {\dot{φ}}_{1} - (p + 1) {‖u_{0}‖}_{σ}^{2} {\dot{φ}}_{1} .

The following task is to claim that J(u) < 0 for t > 0. Arguing by contradiction, we assume that there exist a t₀ > 0 such as $J (u (t_{0})) = 0$ ⁠. Let t₀ > 0 be the first time such as J(u(t)) = 0, thus J(u(t)) < 0 for 0 ≤ t < t₀. From Lemma 5 we obtain ‖u‖_1,2 > z(1) for 0 < t < t₀. Consequently, we get ${‖u (t_{0})‖}_{1,2} \geq z (1)$ and $I (u (t_{0})) \geq h$ which contradicts (3). Then, from (15) we have ${\ddot{φ}}_{1} (t) > 0$ for t > 0. By this and ${\dot{φ}}_{1} (0) = {‖u_{0}‖}_{σ}^{2} \geq 0$ thus there exists a t₀ ≥ 0 such as ${\dot{φ}}_{1} (t_{0}) > 0$ and

φ_{1} (t) \geq {\dot{φ}}_{1} (t_{0}) (t - t_{0}) + φ_{1} (t_{0}) \geq {\dot{φ}}_{1} (t_{0}) (t - t_{0}), t \geq t_{0} .

Then for sufficiently large t we can deduce $(p - 1) φ_{1} > (p + 1) {‖u_{0}‖}_{σ}^{2}$ and

φ_{1} (t) {\ddot{φ}}_{1} (t) - \frac{p + 1}{2} {({\dot{φ}}_{1} (t))}^{2} > 0 .

(17)

Since, for t > 0

{(φ_{1}^{- Θ} (t))}^{''} = - \frac{Θ}{φ_{1}^{Θ + 2} (t)} (φ_{1} (t) \ddot{φ_{1}} (t) - (Θ + 1) \dot{φ_{1}} {(t)}^{2}),

we see that for $Θ = \frac{p - 1}{2}$ we have ${(φ_{1}^{- Θ} (t))}^{''} < 0$ ⁠. Therefore $φ_{1}^{- Θ} (t)$ is concave for sufficiently large t, and there exists a finite time T for which $φ_{1}^{- Θ} (t) \to 0$ ⁠. In other words,

\lim_{t \to T^{-}} φ_{1} (t) = + \infty .

If $0 < I (u_{0}) < h$ ⁠, thus by Lemma 6, we have u(t) ∈ U_λ for 1 < λ < λ₂ and t > 0, where λ₂ is the larger root of equation h(λ) = I(u₀). Therefore J_λ(u) < 0 and (From Lemma 5) ‖u‖_1,2 > z(λ) for λ ∈ (1, λ₂) and t > 0. Then, we have $J_{λ_{2}} (u) \leq 0$ and $‖ u ‖_{1,2} \geq z (λ_{2})$ for t > 0. Thus (15) gives

{\ddot{φ}}_{1} (t) = - 2 J (u) = 2 (λ_{2} - 1) ‖ u ‖_{1,2}^{2} - 2 J_{λ_{2}} (u) \geq 2 (λ_{2} - 1) z^{2} (λ_{2}) > 0, 0 \leq t < \infty, {\dot{φ}}_{1} (t) \geq 2 (λ_{2} - 1) z^{2} (λ_{2}) t + {\dot{φ}}_{1} (0) \geq 2 (λ_{2} - 1) z^{2} (λ_{2}) t, 0 \leq t < \infty, φ_{1} (t) \geq (λ_{2} - 1) z^{2} (λ_{2}) t^{2} + φ_{1} (0) = (λ_{2} - 1) z^{2} (λ_{2}) t^{2}, 0 \leq t < \infty .

Therefore for sufficiently large t we get

\begin{gathered} \frac{1}{2} (p - 1) φ_{1} (t) > (p + 1) {‖u_{0}‖}_{σ}^{2}, \\ \frac{1}{2} (p - 1) {\dot{φ}}_{1} (t) > 2 (p + 1) I (u_{0}) . \end{gathered}

Hence from (16) we again obtain (17) for sufficiently large t. The remainder of the proof is similar to that in the proof of (i).

5. Asymptotic behavior

In this section, we discuss the asymptotic behavior of solutions for problem (1).

Theorem 3.

Let u₀(x) ∈ W^1,2(Ω) and f(u) satisfy (C). Suppose also that $I (u_{0}) < h$ and $J (u_{0}) > 0$ ⁠. Then, for the weak global solution u of problem (1), there exists a constant ω > 0 such as

‖ u ‖_{σ}^{2} \leq {‖u_{0}‖}_{σ}^{2} e^{- ω t}, 0 \leq t < \infty .

(18)

Proof of Theorem 3: By Theorem 1, we know that there exists a global weak solution $u (t) \in L^{\infty} (0, \infty; W^{1,2} (Ω)) \cap C ([0, T]; L^{2} (Ω) \times L^{2} (\partial Ω, ρ))$ to problem (1). Let u(t) be any global weak solution of problem (1) with $I (u_{0}) < h$ and $J (u_{0}) > 0$ ⁠. Consequently,

{〈 \frac{\partial u}{\partial t}, v 〉}_{σ} + 〈 u, v 〉 + 〈 ▽ u, ▽ v 〉 = {〈 f (u), v 〉}_{0}, \forall v \in W^{1,2} (Ω), t \in (0, T) .

(19)

Multiplying (19) by any h(t) ∈ C[0, ∞), we have

{〈 \frac{\partial u}{\partial t}, h (t) v 〉}_{σ} + 〈 u, h (t) v 〉 + 〈 ▽ u, ▽ (h (t) v) 〉 = {〈 f (u), h (t) v 〉}_{0}, \forall v \in W^{1,2} (Ω), t \in (0, T)

and

{〈 \frac{\partial u}{\partial t}, φ 〉}_{σ} + 〈 u, φ 〉 + 〈 ▽ u, ▽ φ 〉 = {〈 f (u), φ 〉}_{0}, \forall φ \in L^{\infty} (0, \infty; W^{1,2} (Ω)), t \in (0, T) .

(20)

Setting φ = u, (20) implies

\frac{1}{2} \frac{d}{d t} ‖ u ‖_{σ}^{2} + J (u) = 0, 0 \leq t < \infty .

(21)

By $0 < I (u_{0}) < h, J (u_{0}) > 0$ and Lemma (4), we get u(t) ∈ S_λ for λ₁ < λ < λ₂ and 0 ≤ t < ∞, where λ₁ < λ₂ are the two roots of equation $h (λ) = I (u_{0})$ ⁠. Consequently, we obtain J_λ(u) ≥ 0 for λ₁ < λ < λ₂ and $J_{λ_{1}} (u) \geq 0$ for 0 ≤ t < ∞. Then, (21) leads to

\frac{1}{2} \frac{d}{d t} ‖ u ‖_{σ}^{2} + (1 - λ_{1}) ‖ u ‖_{1,2}^{2} + J_{λ_{1}} (u) = 0, 0 \leq t < \infty,

accordingly

\frac{1}{2} \frac{d}{d t} ‖ u ‖_{σ}^{2} + (1 - λ_{1}) ‖ u ‖_{σ}^{2} \leq 0, 0 \leq t < \infty .

Finally, Gronwall's inequality, leads to

‖ u ‖_{σ}^{2} \leq {‖u_{0}‖}_{σ}^{2} e^{- 2 (1 - λ_{1}) t}, 0 \leq t < \infty .

This completes the proof of the Theorem.

6. Conclusion

In this manuscript, by using the Galerkin approximation and a family of potential wells, we have proved the existence of a global weak solution of parabolic problem under dynamic boundary conditions (1). In addition, we have studied the blow up and asymptotic behavior of certain weak solutions of problem (1).

The authors would like to thank the referees for their valuable comments and suggestions, which have improved the quality of this paper.

References

Babuska

Osborn

. Eigenvalue problems. In:

Handbook of numerical analysis, finite element method (part I)

, vol.

North-Holland, Amsterdam

;

1991

. p.

641

787

Global well-posedness of solutions to a class of heat equation under dynamical type boundary conditions

1. Introduction

2. Preliminaries

3. Global existence of solutions

4. Blow up in finite time

5. Asymptotic behavior

6. Conclusion

References

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