On a viscoelastic heat equation with logarithmic nonlinearity

. This work deals with the following viscoelastic heat equations with logarithmic nonlinearity In this paper, we show the effects of the viscoelastic term and the logarithmic nonlinearity to the asymptotic behavior of weak solutions. Our results extend the results of Peng and Zhou [ Appl. Anal. 100 (2021), 2804–2824] and Messaoudi [ Progr. Nonlinear Differential Equations Appl. 64 (2005), 351–356.].

In the last several decades, the initial-boundary valued problem to Eq. (1.3) has been studied extensively when the source f (u) is the power functions f (u) = |u| p−2 u, or power like-functions satisfying: (1) f ∈ C 1 and f (0) = f ′ (0) = 0.
(2) (a) f is monotone and is convex for u > 0, and concave for u < 0; or (b) f is convex.
For example, Messaoudi [12] studied Eq. (1.3) in the case f (u) = |u| p−2 u associated with homogeneous Dirichlet boundary condition. By the convexity method, the author showed that if the relaxation function g is non-negative and non-increasing satisfying ∞ 0 g(s)ds < 2(p − 2) 2p − 3 , then weak solution to (1.3) blows up in finite time provided initial energy is positive. In [20], Truong and Y also studied the problem of the above type with f (u) in the general polynomial type and they obtained the existence, blow up and asymptotic behavior for weak solution under suitable conditions. For further results on the existence, blow-up or asymptotic behavior of solutions, we refer the reader to [5,13,16,19] in case of power or power-like sources.
With regard to the logarithmic nonlinearity, there are a few results (see [1,2,7,9,15]). In case the relaxation function g vanishes, the problem (1.1) reduces to the following: in Ω. (1.4) In case p = 2, self-similar solutions and their asymptotic stability for (1.4) 1 has been studied by Samarskii et al. [17]. With regard to weak solutions, by using the potential well method and the logarithmic Sobolev inequality in H 1 0 (Ω) (see [6,11]), Chen et al. [1] prove that the weak solution blows up at infinite time and exists globally provided that the initial data start in the stable sets and unstable sets respectively. This result is so interesting because it showed the different effect of logarithmic nonlinearity compared to the power one. Inspired by this result the second and third authors [9] extended (1.4) to the evolution p-Laplacian equations and showed a different result compared to the case p = 2, confirming that weak solutions blow up in finite time. Afterward the PDEs with logarithmic nonlinearity have been attracted many researchers, see [2,7,15] for example. In particular, Peng and Zhou [15] have showed recently that in case p > 2 the solutions of (1.4) behave like the nonlinear case f (u) = |u| p−2 u. These results shows that p = 2 is the critical exponent for the blow-up at infinite time.
Motivated by all these works, our aim in this paper is to study the effect of the viscoelastic term t 0 g(t − s)∆u(s)ds and the logarithmic nonlinearity |u| p−2 u ln |u| to the blow-up and global existence of weak solutions to (1.1). Firstly, the presence of logarithmic nonlinearity help us relax conditions on g compared to [12], that is, Secondly, because of the presence of t 0 g(t − s)∆u(s)ds we need more restriction on the range of p and for small energy levels E(0) < d δ ≤ d (see (2.2) below) compared to [15].
Our result is twofold in the sense that it is not only study the blow-up in finite time but also global existence of weak solutions. In addition, we also give the lower and upper bound for blow-up time and decay estimate of global solutions. Also notice that our method differs from [12]. To obtain the main results, we employ the ideas from the potential well method due to Sattinger [18] (see also [14]). However, since the presence of the relaxation g we could not apply the stable and unstable sets as in [14]. To overcome this difficulty we construct a family of potential wells (see (2.3) and (2.4)) that is more suitable for the PDEs involving viscoelastic terms. Also notice that the asymptotic behavior of global solutions in [15] has not been studied and it can be done by using the method employed in this paper. This paper is organized as follows. In the next section, we present some preliminaries and define the family of modified potential wells. Our main results are stated in the Section 3 and the rest of the paper is devoted to their proofs.

Preliminary lemmas
The following lemmas will be needed in our proof of the main results.
is compact.

Lemma 2.3 ([9]
). Let ρ be a positive number. Then we have the following elementary inequalities: is a positive function satisfying the following inequality

Modified potential wells
For 0 < δ ≤ ℓ with ℓ := 1 − ∞ 0 g(s)ds, we define potential energy functional and the associated Nehari functional then we have that We have the following lemma.
The last statement (iii) follows from (i)-(ii) and the relation The proof is complete.
Let us state here the Sobolev imbedding which can be found in [4].
Lemma 2.6. Assume that p is a constant such that where p ∈ [1, ∞) can be any constant. Then H 1 0 (Ω) → L p (Ω) continuously, and there exists a positive constant C p depending on n, p and Ω such that ∥u∥ p ≤ C p ∥∇u∥ holds for all u ∈ H 1 0 (Ω). We choose C p be the optimal constant satisfying the above inequality, i.e.
It follows that The conclusions then follow from the above inequality.
Let us define the so-called Nehari manifold associated to the energy functional J δ by By Lemma 2.5 we know that N δ is not empty set. It is clear that J δ (u) is coercive on the Nehari manifold N δ , hence we can define The standard variational method shows that d δ is a positive finite number and therefore it is well-defined.
We end this section by giving the definitions of the modified stable and unstable sets as in [14].

Main results
Throughout this paper, we make the following usual assumptions on the relaxation function g: (G) g : R + → R + belongs to C 1 (R + ) and satisfies the conditions Let us now give the definition of weak solutions to (1.1).
Proof. By substituting φ = u t in (3.1), we get after some simple calculations that Then, using the assumption (G, (i)), it follows that E(t) is an non-increasing functional and satisfies the energy inequality The proof is complete.
We are now in the position to state the main theorems of this paper.

Proof of Theorem 3.3
Based on the Faedo-Galerkin method, this proof consists of three steps.
Step 1. Finite-dimensional approximations. Let {w j } be the orthogonal complete system of eigenfunctions of −∆ in H 1 0 (Ω) , which is orthonormal in L 2 (Ω). We find the approximate solution of the problem (1.1) in the forms where the coefficients functions c mj , 1 ≤ j ≤ m, satisfy the system of integro-differential equations u mt , w j + ∇u m , ∇w j −   Step 2. A priori estimate. Multiplying (4.2) by c ′ mj (t) and summing for j from 1 to m, we get ⟨u mt , u mt ⟩ + ⟨∇u m , ∇u mt ⟩ − From E(0) < d δ and (4.3), we deduce that E m (0) < d δ for sufficiently large m. And then, we deduce from (4.6) and (4.7) that holds for sufficiently large m. Take note of I δ (u 0 ) > 0, we can conclude that u 0 ∈ W δ . It implies from (4.3) that u m (0) ∈ W δ for sufficiently large m. Now, we will show that u m (t) ∈ W δ for any t ∈ [0, T m ] and sufficiently large m. In fact, if not, there exists a t 0 ∈ (0, T m ] and a sufficient large m such that I δ (u m (t 0 )) = 0 and u m (t 0 ) ̸ = 0, then we get that u m (t 0 ) ∈ N δ . So we deduce from the definition of d δ that J δ (u m (t 0 )) ≥ d δ , which contradicts (4.8). Thus, u m (t) ∈ W δ for any t ∈ [0, T m ] and sufficient large m, which implies I δ (u m (t)) ≥ 0 for any t ∈ [0, T m ] and sufficient large m.
On the other hand, by (4.10), we get , and S q is the best constant of the Sobolev embedding H 1 0 (Ω) → L q (Ω).

Proof of Theorem 3.4
We begin this section by the following useful lemma which is useful later on.
We now divide the proof of the Theorem 3.4 into two following steps: Step 1: Blow-up in finite time and upper bound estimate of the blow-up time.
By contradiction, we assume that u(t) exists globally and define the function where b and T 0 are positive constants to be determined later. Then we have and By using (1.1), we deduce from (5.4) that On the other hand, by the Hölder inequality and the Cauchy-Schwarz inequality, we have and by the Young inequality, one has It follows from (5.2)-(5.7) that where ζ : [0, T] → R is the function defined by On the other hand, from (3.2) we have that Since b satisfies (5.13), by minimizing the above inequality for T 0 > 2∥u 0 ∥ 2 (p−2)b , we arrive at Step 2: Lower bound estimate of the blow up time.
Recalling the Lemma 3.2, we have Let us divide Ω into two parts as follows: Applying Lemma 2.3, Hölder's inequality, Young's inequality, we reach Here, for simplicity, we write u instead of u(t).

Proof of Theorem 3.5
We begin with the following lemma which is helpful to the proof of Theorem 3.5.
Lemma 6.1. Under the assumptions of the Theorem 3.3. For any 0 < δ ≤ ℓ, we have that Proof. It is first noticed that u 0 ∈ W δ thanks to E(0) < d δ and I(u 0 ) > 0. By using the similar method as in the proof of Lemma 5.1, we can show that u(t) ∈ W δ for t ≥ 0. Taking this into account and using the Lemma 2.5 (iii), we imply that there is a constant λ 1 > 1 such that I δ (λ 1 u(t)) = 0.
Proof. By virtue of Lemma 6.1 and the definition of E(t), we have that Taking this into account, we deduce from the definition of ρ(t) that where S 2 is the optimal constant in the embedding H 1 0 (Ω) → L 2 (Ω). From (G, iii) we have ξ(t) ≤ ξ(0) ≤ M for some constant M > 0. Combining with the above estimate to obtain By choosing ε small such that 0 < ε < 1/C (M) we claim the lemma.
Since E(0) < ℓ 2δ p p−2 d δ , we can pick 0 < Λ < p such that Therefore, we get which implies This completes the proof.