Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time

1 Introduction

In this study, we consider the following nonlinear parabolic equation:

associated with the homogeneous Robin boundary conditions

and supplemented with initial condition

where h 0 , h 1 ⩾ 0 are given constants such that h 0 + h 1 > 0 , and μ 1 , μ 2 , g , f , u 0 are given functions satisfying conditions to be specified later.

Problem (1.1)–(1.3) without viscoelastic term and μ 1 = 1 is called heat equation or parabolic equation and has the simple form

Problem (1.1)–(1.3) along with (1.4), together with appropriate boundary and initial conditions, arises naturally in engineering and physical sciences. Such problems have been extensively investigated, and numerous results regarding existence, nonexistence, regularity, and asymptotic behavior have garnered significant attention from mathematicians. For instance, Gazzola and Weth [1] considered the following initial-boundary value problem

where Ω ⊂ R n is an open bounded domain with smooth boundary ∂ Ω , and 1 < p < n + 2 n − 2 . By using potential well method, which was first introduced by Payne and Sattinger [2], the authors studied the dichotomy between global existence and blowup for the resulting solutions via the initial energy. Roughly speaking, the natural phase space can be divided into three parts corresponding to three initial energy levels. Since the initial energy level is over the mountain pass level, by exploiting comparison principle as well as strong parabolic maximum principle, the authors gave us some properties of the resulting solution. Inspired by this celebrated research, a vast number of research works related to parabolic equations, coupled parabolic equation or pseudo-parabolic equation have been investigated. Interested readers can find it in [3–10]. The common point of these studies is that the authors just consider the equations containing the coefficient independence on spatial-time variables. Problem (1.1)–(1.3) n is the mathematical model of many natural phenomena in physical science and engineering. For example, in the study of heat conduction in materials with memory, to our best knowledge, the first study that treats the heat equation with coefficient dependence on spatial-time variables is [11]. In this study, the authors consider the following nonlinear heat equation:

associated with the homogeneous Robin boundary conditions

and supplemented with initial condition.

First, by adopting the Faedo-Galerkin method, the authors established the existence and uniqueness of the weak solutions. Subsequently, by employing certain differential inequalities, the authors provided a sufficient condition for finite-time blowup as well as exponential decay estimates. With the inclusion of the term μ 1 and the viscoelastic term ∫ 0 t g ( t − s ) ∂ ∂ x ( μ 2 ( x , s ) u x ( x , s ) ) d s in our equation, we cannot employ the potential well method to identify invariant sets under the flow of our problem. Consequently, obtaining a priori estimates for the solution becomes challenging. To address this issue, we need to utilize an alternative method first introduced by Vitillaro [12]. This method has been widely employed to investigate problems involving viscoelastic terms, as in [13–15]. However, some open questions remain related to this problem. These questions can be stated as follows:

1. In order to prove the blowup results, the authors just focused on the negative initial energy, i.e., E ( 0 ) < 0 . Thus, a natural question arises: can we prove that the solution blows up in finite time with nonnegative initial energy?

2. In order to obtain the decay estimate for solution at infinity, the authors required that the relaxation function g satisfy the inequality

   g ′ ( t ) ⩽ − ξ 1 g ( t ) , ∀ t ∈ [ 0 , ∞ ) ,

   where ξ 1 is a positive constant. This estimate, combined with the positively of the function g , yields g ( t ) ≲ exp ( − ξ 1 t ) . This means that the relaxation function g must be dominated by an exponential function. Thus, a natural question arises: can we consider the generalized case

   g ′ ( t ) ⩽ − ξ ( t ) g ( t ) , ∀ t ∈ [ 0 , ∞ ) .

1. In Section 2, we introduce some notations, preliminaries, and present the local Hadamard well-posedness result.

2. In Section 3, we introduce a novel method for proving the blowup result with both negative and nonnegative initial energy. The innovation in this section lies in our ability to address both cases simultaneously using the same auxiliary function;

3. Section 4 is dedicated to proving a sufficient condition for the global existence and decay of weak solutions.

2 Preliminary results and notations

For the sake of convenience, we put Ω = ( 0,1 ) . We omit the definitions of the usual function spaces such that L p = L p ( Ω ) and H m = H m ( Ω ) . Let ⟨ ⋅ , ⋅ ⟩ be the scalar product in L 2 and ⟨ ⋅ , ⋅ ⟩ H 1 be the scalar product in H 1 . The notation ∥ ⋅ ∥ p stands for the L p norm, ∥ ⋅ ∥ = ∥ ⋅ ∥ 2 , and ∥ ⋅ ∥ X for the norm in the Banach space X . We denote by X ′ the dual space of X . We denote by L p ( 0 , T ; X ) with p ∈ [ 1 , ∞ ] for the Banach space of measurable functions u : ( 0 , T ) ⟶ R , such that

and

Let μ 1 , μ 2 ∈ C ( Ω ¯ × [ 0 , T ] ) such that μ i ( x , t ) ⩾ μ ̲ i > 0 for all ( x , t ) ∈ Ω × [ 0 , T ] and for all i ∈ { 1 , 2 } . We consider two families of symmetric bilinear forms { a i ( t ; ⋅ , ⋅ ) } 0 ⩽ t ⩽ T on H 1 × H 1 defined by

for all u , v ∈ H 1 , t ∈ [ 0 , T ] , and i ∈ { 1 , 2 } . We have the following lemmas. The proofs are straightforward, so we omit the details.

We conclude this section by presenting the local Hadamard well-posedness results. To achieve this, we first provide a precise definition of the weak solution to problem (1.1)–(1.3).

To ensure the existence results, we must stipulate the following hypotheses:

1. u 0 ∈ H 1 ;

2. μ 1 , μ 1 ′ ∈ C ( Ω ¯ × [ 0 , ∞ ) ) and there exists the constant μ ̲ 1 such that μ 1 ( x , t ) ⩾ μ ̲ 1 > 0 for all ( x , t ) ∈ Ω ¯ × [ 0 , ∞ ) ;

3. μ 2 ∈ C ( Ω ¯ × [ 0 , ∞ ) ) ;

4. f ∈ C ( Ω ¯ × [ 0 , ∞ ) × R ) ;

5. g ∈ C 1 ( [ 0 , ∞ ) ) .

3 Finite time blowup and lifespan estimates

The main goal of this section is to show that with some suitable conditions, every weak solution to problem (1.1)–(1.3) blows up in finite time. In this section, we aim to consider problem (1.1)–(1.3) in a specific case f ( x , t , u ) = K ( x , t ) f ( u ) , μ 2 ( x , t ) = μ 2 ( x ) . First, we posit the following assumptions:

1. μ 1 , μ 1 ′ ∈ C ( Ω ¯ × [ 0 , ∞ ) ) , and there exists the constant μ ̲ 1 such that μ 1 ( x , t ) ⩾ μ ̲ 1 > 0 , and ∂ μ 1 ∂ t ( x , t ) ⩽ 0 for all ( x , t ) ∈ Ω ¯ × [ 0 , ∞ ) ;

2. μ 2 ∈ C ( Ω ¯ ) and there exists the constant μ ̲ 2 such that μ 2 ( x ) ⩾ μ ̲ 2 > 0 for all x ∈ Ω ¯ .

Now, on the product space H 1 × H 1 , we consider the following symmetric bilinear forms:

Furthermore, it is easy to prove that the forms a ( ⋅ , ⋅ ) , b ( ⋅ , ⋅ ) are continuous on H 1 × H 1 and coercive on H 1 . On the other hand, the norms v ⟼ ∥ v ∥ H 1 , v ⟼ ∥ v ∥ a = a ( v , v ) , and v ⟼ ∥ v ∥ b = b ( v , v ) are equivalent. In fact, we have the following lemmas.

To derive blowup results, we also need to specify the following set of assumptions:

1. K , K t ∈ C ( Ω ¯ × [ 0 , ∞ ) ) satisfying

   1. 0 ⩽ K ( x , t ) for all ( x , t ) ∈ Ω ¯ × [ 0 , ∞ ) ;

   2. K t ( x , t ) ⩾ 0 for all ( x , t ) ∈ Ω ¯ × [ 0 , ∞ ) .

2. f ∈ C 1 ( R ) and there exists a constant p > 2 such that

   u f ( u ) ⩾ p ∫ 0 u f ( z ) d z = p F ( u ) ⩾ 0 , ∀ u ∈ R .

3. g ∈ C 1 ( [ 0 , ∞ ) ) with the following properties:

   1. g ( t ) > 0 for all t ∈ [ 0 , ∞ ) ;

   2. g ′ ( t ) ⩽ 0 for all t ∈ [ 0 , ∞ ) ;

   3. G ∞ ⩽ p ( p − 2 ) ( p − 1 ) 2 μ ̲ 1 μ ¯ 2 with G ∞ = lim t → ∞ G ( t ) , where G ( t ) = ∫ 0 t g ( s ) d s .

where

In the next theorem, we will prove that the weak solution u blows up in finite time, provided the initial energy is negative.

Next we state the blowup result when the initial energy is nonnegative and small enough. In this case, we make the following assumptions:

1. K , K t ∈ C ( Ω ¯ × [ 0 , ∞ ) ) such that

   1. 0 ⩽ K ( x , t ) ⩽ K 2 for all ( x , t ) ∈ Ω ¯ × [ 0 , ∞ ) with K 2 being a positive constant;

   2. K t ( x , t ) ⩾ 0 for all ( x , t ) ∈ Ω ¯ × [ 0 , ∞ ) .

2. f ∈ C 1 ( R ) and there exist constants d ¯ 2 > 0 , d 2 > p > 2 , q i > p for all i ∈ { 1 , … , N } such that

   1. u f ( u ) ⩾ p ∫ 0 u f ( z ) d z = p F ( u ) ⩾ 0 for all u ∈ R ;

   2. u f ( u ) ⩽ d 2 F ( u ) ⩽ d ¯ 2 d 2 ∣ u ∣ p + ∑ i = 1 N ∣ u ∣ q i for all u ∈ R .

First, we put