Nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case

In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow-up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $11$ for $n=1$. Here the Strauss exponent $p_{\mathrm{Str}}(n)$ is the critical exponent for the semilinear wave equation with power nonlinearity.


Introduction
In recent years, the Moore -Gibson -Thompson (MGT) equation, a linearization of a model for wave propagation in viscous thermally relaxing fluids, has caught a lot of attention (see [29,39,14,20,19,28,18,25,31,3,6,24,7,23,30,2,5] and references therein). This model is realized through the third order hyperbolic partial differential equation τ u ttt + u tt − c 2 ∆u − b∆u t = 0. (1.1) In the physical context of acoustic waves, the unknown function u = u(t, x) denotes a scalar acoustic velocity, c denotes the speed of sound and τ denotes the thermal relaxation. Besides, the coefficient b = βc 2 is related to the diffusivity of the sound with β ∈ (0, τ ]. In particular, there is a transition from a linear model that can be described with an exponentially stable strongly continuous semigroup in the case 0 < β < τ to the limit case β = τ , where the exponential stability of a semigroup is lost and it holds the conservation of a suitable defined energy (see [20,28]). For this reason, we shall call the limit case β = τ the conservative case.
In this paper, we consider the semilinear Cauchy problem associated to the MGT equation βu ttt + u tt − ∆u − β∆u t = |u| p , x ∈ R n , t > 0, (u, u t , u tt )(0, x) = (u 0 , u 1 , u 2 )(x), x ∈ R n , (1.2) in the limit case τ = β > 0, where p > 1 and, for the sake of simplicity, we normalized the speed of the sound by putting c 2 = 1. We are interested to the blow -up in finite time of local (in time) solutions under suitable sign assumptions for the Cauchy data regardless of their size and for suitable values of the exponent p.
The lifespan of solutions to (1.3) that blow up in finite time has been intensively considered. Here we refer to [26,41,42,43,27,9,37,46,36,15]. According to the works listed above, the sharp estimates for the lifespan T (ε) are given by for n 3 and for n = 2 and 2 < p < p Str (2), by for n = 2 and 1 < p 2, where a = a(ε) satisfies a 2 ε 2 log(1 + a) = 1, and by for n = 1 and p > 1, where in each case ε is a sufficiently small positive quantity. Note that, for the sake of simplicity, in the previous lifespan estimates for low dimensions n = 1, 2 we restricted our considerations to the case in which the integral of u 1 is not zero. Our main result Theorem 4.1, which is stated and proved in Section 4, is a blow -up result for the semilinear model (1.2) that holds for exponents of the nonlinearity such that 1 < p < p Str (n) and under suitable sign assumptions for compactly supported initial data. Furthermore, we will obtain an upper bound estimate for the lifespan of local solutions to (1.2) which coincides in some cases with the optimal one for (1.3), as we have just recalled. The proof of Theorem 4.1 is based on an iteration argument, which allows us to show the blow -up in finite time of the space average of a local in time solution to (1.2).
Let us point out that this iteration argument is not just a straightforward generalization of the one for (1.3). Indeed, in the iteration procedure we have to deal with an unbounded exponential multiplier. For this purpose, we propose a slicing procedure of the domain of integration by taking inspiration from [1], even though the sequence of the parameters (cf. {L j } j∈N below in Subsection 4.3), that characterize the slicing of the domain of integration, has a completely different structure. Up to our best knowledge, our result is the first attempt to include an unbounded exponential multiplier in an iteration argument for proving a blow -up result for hyperbolic semilinear models.
The present paper is organized as follows. In Section 2 we first derive L 2 -L 2 estimates and well -posedness for the linear MGT equation. In Section 3 combining Banach fixed point theorem with the derived L 2 -L 2 estimates, the local (in time) existence of solutions to the semilinear MGT equation is proved. Then, in Section 4 we apply an iteration method associated with the test function introduced in [40] to prove the blow -up of energy solutions in the sub -Strauss case. Finally, some concluding remarks in Section 5 complete the paper.
Notation: We give some notations to be used in this paper. We write f g when there exists a positive constant C such that f Cg. We denote g f g by f ≈ g. Moreover, B R denotes the ball around the origin with radius R in R n . As mentioned in the introduction, p Str (n) denotes the Strauss exponent.

Linear problem for the MGT equation
In this section, we will derive some qualitative properties of solutions to the corresponding linearized Cauchy problem to (1.2), which is advantageous for us in order to understand the semilinear problem. More precisely, we are interested in the following linear MGT equation: where β is a positive constant. According to [20,24], a suitable defined energy for the MGT equation is conserved from the point of view of semigroups.
We now state the energy conservation result for the linear homogeneous Cauchy problem.
The conservation of the energy stated in Proposition 2.1 leads immediately to the uniqueness of the solution to the Cauchy problem (2.4).
To conclude this section, we point out that the solution to the linear Cauchy problem for MGT equation (2.4) fulfills the inhomogeneous wave equation Thus, we claim that supp u(t, ·) ⊂ B R+t , if we assume supp u j ⊂ B R for any j = 0, 1, 2 and for some ) has support contained in the forward cone {(t, x) : |x| R + t} under these assumptions and we can use the property of finite speed of propagation for the classical wave equation.

Existence of local (in time) solution
for any j = 0, 1, 2 and for some R > 0. We assume p > 1 such that p n/(n − 2) when n 3. Then, there exists a positive T and a uniquely determined local (in time) mild solution Let us introduce some notations for the proof of the local (in time) existence of solutions. We denote by K 0 (t, x), K 1 (t, x) and K 2 (t, x) the fundamental solutions to the linear Cauchy problem (2.4) with initial data (u 0 , u 1 , u 2 ) = (δ 0 , 0, 0), (u 0 , u 1 , u 2 ) = (0, δ 0 , 0) and (u 0 , u 1 , u 2 ) = (0, 0, δ 0 ), respectively. Here δ 0 is the Dirac distribution in x = 0 with respect to spatial variables. Therefore, the solution to (2.4) is given by where the Fourier transforms of the kernels K 0 (t, x), K 1 (t, x) and K 2 (t, x) are given by Proof. Let us define the family of evolution spaces with the norm .
According to Duhamel's principle, we introduce the operator We will consider as mild local in time solutions to (1.2) the fixed points of the operator N . Therefore, with the aim of deriving the local (in time) existence and uniqueness of the solution in X(T ), we need to prove First of all, from Proposition 2.2 it is clear that and u ln Next, to prove (3.7), we apply the classical Gagliardo -Nirenberg inequality. Thus, we get for Then, applying the previous inequality and using L 2 -L 2 estimates from Proposition 2.2, we derive Ct u p X(T ) , for any ℓ, j ∈ N 0 such that 1 ℓ + j 2.
Finally, N u satisfies the support condition supp N u(t, ·) ⊂ B R+t for any t ∈ [0, T ], since w = N u is a solution of the inhomogeneous Cauchy problem for the wave equation and u is supported in the forward cone due to u ∈ X(T ). Thus, we may conclude that N maps X(T ) into itself and (3.7). To derive (3.8), we remark that .
By employing and Hölder's inequality, we conclude . Finally, by using L 2 -L 2 estimates from Proposition 2.2 again, we immediately obtain the desired estimate (3.8). This completes the proof.
where ε > 0 is a parameter describing the smallness of initial data. The aim of this section is to prove the blow -up of local (in time) solutions to (4.9) in the sub -Strauss case, that is for 1 < p < p Str (n), under suitable conditions for the Cauchy data, and to derive an upper bound estimate for the lifespan. To do this, we introduce first the definition of energy solutions to (4.9).
We say u is an energy solution to (4.9) for any ψ ∈ C ∞ 0 ([0, T ) × R n ) and any t ∈ [0, T ). Applying a further step of integration by parts in (4.10), it results In particular, letting t → T , we find that u fulfills the definition of weak solution to (4.9).
Theorem 4.1. Let us consider p > 1 such that Let (u 0 , u 1 , u 2 ) ∈ H 2 (R n ) × H 1 (R n ) × L 2 (R n ) be nonnegative and compactly supported functions with supports contained in B R for some R > 0 such that u 0 is not identically zero. Let be an energy solution on [0, T ) to the Cauchy problem (4.9) according to Definition 4.1 with lifespan T = T (ε) such that supp u(t, ·) ⊂ B R+t for any t ∈ (0, T ). (4.12) Then, there exists a positive constant ε 0 = ε 0 (u 0 , u 1 , u 2 , n, p, R, β) such that for any ε ∈ (0, ε 0 ] the solution u blows up in finite time. Furthermore, the upper bound estimate for the lifespan holds, where C is an independent of ε, positive constant and θ(p, n) . = 2 + (n + 1)p − (n − 1)p 2 . (4.13)

Iteration frame
According to Theorem 4.1, we assume that u 0 , u 1 and u 2 are nonnegative functions, with nontrivial u 0 , and compactly supported with support contained in B R for some suitable R > 0. Then, thanks to what we underlined in Section 3, we have supp u(t, ·) ⊂ B R+t for any t ∈ (0, T ). (4.14) We introduce now the following time -dependent functional: Choosing a test function ψ in (4.10) such that ψ = 1 on {(s, x) ∈ [0, t] × R n : |x| R + s}, due to (4.14) we have Differentiating the previous relation with respect to t, we get By employing Hölder's inequality and (4.14), we may estimate where C = C(n, p) > 0 is a constant that depends on the measure of the unitary ball. Hence, from (4.15) we obtain The previous ordinary differential inequality for F allows us to the derive the frame for our iteration argument. In other words, integrating twice, by (4.16) we have Then, multiplying the last inequality by e s/β and integrating over [0, t], we arrive at From the integral inequality (4.17) we have a twofold consequence. Since we assume that initial data are nonnegative, then (4.17) implies F (t) ε for any t 0. So, in particular, F is a positive function.
On the other hand, if we neglect the terms involving F (0), F ′ (0) and F ′′ (0) in (4.17), then, we find (4.18) We point out explicitly that (4.18) will play a fundamental role in our iteration argument: this is, in fact, the frame which allows us to determine a sequence of lower bound estimates for the function F .

Lower bound for the functional
Even though we proved that F (t) ε in the last subsection, this lower bound for F is too weak in order to start with the iteration procedure. For this reason we will improve this lower bound for F by introducing a second time -dependent functional. Let us consider the function This function has been introduced for the first time in the study of blow -up results for wave models in [40]. The function Φ is a positive smooth function that satisfies the following crucial properties: Furthermore, we introduce the function with separate variables Ψ = Ψ(t, x) = e −t Φ(x). Clearly, Ψ is a solution of the adjoint equation to the homogeneous linear MGT equation, namely, We can introduce now the definition of the second functional F 1 as follows: Since Ψ is a positive function, applying (4.11) with test function Ψ, we get , where in the second last step we used that Ψ solves (4.21), while in the last step we used the obvious representations Note that we may employ Ψ as test function even though it has no compact support thanks to the support property for u in (4.12). As outcome of the previous chain of equalities, we get that We can rewrite the left -hand side of (4.22) as while the right -hand side depends only on initial data Multiplying (4.22) by e (1+1/β)t and integrating over [0, t], we get Analogously to the last step, we multiply the previous inequality by e 2t and we integrate over [0, t], so that for β = 1, while for β = 1 we get 4F 1 (0)) . Therefore, thanks to the assumptions on u 0 , u 1 , u 2 , the previous estimates yield easily (4.23) where the unexpressed multiplicative constant depends on u 0 , u 1 , u 2 . Let us show now how (4.23) provides a lower bound estimate for the spatial integral of the nonlinearity |u| p . Applying Hölder's inequality, we have where p ′ denotes the conjugate exponent of p. In the literature, it is well -known that the p ′ power of the L p ′ (B R+T ) -norm of Ψ(t, ·) can be estimate in the following way: Finally, we combine (4.15) and (4.24) in order to get a lower bound for F which will allow us to start with the iteration argument. Repeating the same intermediate steps that we did in order to prove (4.18) starting from (4.16), from (4.15) we find Next, we plug the lower bound (4.24) in the last estimate. Thus, we obtain In particular, for t β the factor containing the exponential function in the last line of the previous chain of inequalities can be estimate from below by a constant, namely, for any t β, (4.25) where the multiplicative constant is = n + 1.

Iteration argument
In the previous subsection, we derived a first lower bound for F . Now we will derive a sequence of lower bounds for F by using the iteration frame (4.18). More precisely, we will show that where {C j } j∈N , {α j } j∈N and {γ j } j∈N are sequences of nonnegative real numbers that we will determine throughout the proof and {L j } j∈N is the sequence of the partial products of the convergent infinite product that is, ℓ k for any j ∈ N.
Note that (4.25) implies (4.26) for j = 0. We are going to prove (4.26) by using an inductive argument. Therefore, it remains to prove just the inductive step. Let us assume the validity of (4.26) for j 0. We will prove (4.26) for j + 1. After shrinking the domain of integration in (4.18), if we plug (4.26) in (4.18), we get for t L j+1 β. Note that in the last step we could restrict the domain of integration with respect to s from [L j β, t] to [t/ℓ j+1 , t] because t L j+1 β and ℓ j+1 > 1 imply L j β t/ℓ j+1 < t. Consequently, for t L j+1 β. Finally, we remark that for t L j+1 β ℓ j+1 β we may estimate Also, for t L j+1 β we have shown which is exactly (4.26) for j + 1, provided that

Concluding remarks
Let us consider the general case of the Cauchy problem for the semilinear MGT equation τ u ttt + u tt − ∆u − β∆u t = |u| p , x ∈ R n , t > 0, (u, u t , u tt )(0, x) = (u 0 , u 1 , u 2 )(x), x ∈ R n , (5.32) where 0 < τ < β (the dissipative case) or τ = β (the conservative case) and p > 1. By introducing w . = τ u t + u, we may transform (5.32) to the following semilinear second order evolution equation: w tt − β τ ∆w + β−τ τ 2 G * ∆w = H(u 0 , w; p, β, τ ), x ∈ R n , t > 0, (w, w t )(0, x) = (τ u 1 + u 0 , τ u 2 + u 1 )(x), x ∈ R n , (5.33) where the kernel function G = G(t) is given by G(t) . = e −t/τ and the right -hand side is H(u 0 , w; p, β, τ )(t, x) . = − β−τ τ e −t/τ ∆u 0 (x) + e −t/τ u 0 (x) Here the convolution term is defined by We now may understand the equation in (5.33) in the two cases above mentioned. In the conservative case τ = β, we may interpret the model (5.33) as a wave equation with power source nonlinearity, which includes a memory term with an exponential decaying kernel function. On the other hand, in the dissipative case 0 < τ < β, the dissipation generated by the memory term comes into play even in the linear part, therefore, the model (5.33) can be interpreted as a semilinear viscoelastic equation (see for example [8,32]). Up to the knowledge of the authors, the blow -up of solutions to this kind of semilinear viscoelastic equations is still an open problem. In this paper, we considered the Cauchy problem for the semilinear MGT equation with power nonlinearity |u| p and proved the blow -up of local energy solutions in the sub -Strauss case, i.e., for 1 < p < p Str (n). Concerning the Cauchy problem for the semilinear MGT equation with nonlinearity of derivative type in the conservative case, namely, βu ttt + u tt − ∆u − β∆u t = |u t | p , x ∈ R n , t > 0, (u, u t , u tt )(0, x) = (u 0 , u 1 , u 2 )(x), x ∈ R n , (5.34) with β > 0, in the forthcoming paper [4], we shall study the blow -up of local in time solutions to (5.34) and the corresponding lifespan estimates under suitable assumptions for initial data. More specifically, the blow -up in finite time of energy solutions to (5.34) is going to be proved providing that the power p of the nonlinearity satisfies 1 < p p Gla (n) . = n + 1 n − 1 , for n 2 and p > 1 for n = 1. We underline that the Glassey exponent p Gla (n) is the critical exponent for the corresponding semilinear wave equation with nonlinearity of derivative type.