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We investigate the large time behavior for the inhomogeneous damped wave equation with nonlinear memory , imposing the condition where , , , , , and . Namely, it is shown that, if , and , then for all , the considered problem has no global weak solution.
We study the Cauchy problem of nonlinear damped wave equation in the form
imposing the initial condition
where , , , with , and . The integral in the right-hand side of Equation (1) is known as “nonlinear memory”.
We recall that the choices of both the domain and boundary conditions may influence significantly the properties and behavior of the physical system, which is mathematically represented by the above Equation (1). In general, the wave type equations are fundamental tools in recasting various propagation phenomena and in developing methods for numerical solving the physics problems. In detail, symmetries of wave type equations and their solutions have been pointed out and investigated by many contributors. We mention that the symmetry’s properties were successfully used to obtain ortogonality’s criteria for the existence of solutions in elastic and anisotropic media (see, for example, the pioneering papers of Love , Woodhouse , Chapman–Woodhouse , and the references therein). About the computational approach to the study of wave type equations, we recall that the nonlinear wave equations can be linked to linear wave equations, using symmetry transformations (in particular, non-local transformations). So, we can find a correspondence one-to-one between the solutions of nonlinear and linear wave equations. Resuming, each nonlinear wave equation can be linearized by a non-local symmetry analysis (for more details, see Bluman–Cheviakov ). For further discussion about the potential benefit of this procedure, we refer to the papers by Taylor–Kidder–Teukolsky  and Palacz  (spectral methods for propagation phenomena).
Here, we ask the question of whether the problem (1) and (2) admits global weak solutions. The interest in such a kind of results is motivated by a wide literature on what can be called “large time behavior of solutions” to wave type problems, which consists in providing sufficient criteria to the existence, nonexistence and blow-up of solutions to some classes of parabolic differential equations. For this purpose, we employ the test function method.
Now, we recall some important results related to the blow-up of solutions to damped wave equations. First, we refer to the semilinear damped wave equation
imposing (2). In the work of Todorova–Yordanov , can be found the following results:
If and , , then there is no global solution to problem (2) and (3);
If for , and for , then a unique global solution exists, under suitable initial values.
In the literature, the exponent “” is known as the critical Fujita exponent. Indeed, it is critical for the problem (2) and (3), but it is also the critical exponent for the semilinear heat equation (see Fujita )
Further studies in Kirane–Qafsaoui  (semi-linear wave equation with linear damping) and Zhang  (nonlinear wave equation with damping), established that belongs to the nonexistence case.
It is worth pointing out that in the limit case , (1) reduces to
The above equation was recently investigated by Jleli–Samet , who established the following results:
If , , , for , and for , then there is no global weak solution to (4);
If and , then global solutions exist for suitable .
This means that the critical exponent for Equation (4) is given by
From the above result, one observes the considerable effect of the inhomogeneous term μ on the critical behavior of (2) and (3). Namely, for , the critical exponent for (4) jumps from (the critical exponent for (2) and (3)) to the bigger exponent . Notice that a similar phenomenon was observed for the heat equation  (Zhang, 1998) and the wave equation  (Zhang, 2000).
An interesting wave type problem is driven by the equation:
assuming the initial condition (2). Problem (2) and (5) was first investigated by Fino , who proved the results as follows. Let
Thus, we have:
If , then , for all ;
If , then
Moreover, a finite time blow-up occurs in the following cases (see again ):
If and , i.e., for all when , or for all when ;
If and , i.e., and .
Finally, by considering compactly supported functions with small values, in  a global existence result is derived in the case and , where
Studying the same problem (2) and (5), D’Abbicco  obtained a global existence result for , where and , imposing suitable initial conditions.
The previous contributors give motivation to our work here. Indeed, we aim to study the effect of the inhomogeneous term on the large time behavior for problem (2) and (5).
Under sufficient conditions on the inhomogeneous term and the functions , , a nonexistence result is given in the following main result.
then problem (1) and (2) admits no global weak solution, for all .
As a byproduct of Theorem 1, one deduces that the critical exponent for problem (1) and (2) is equal to ∞, for all .
In the next Section 2 we collect the auxiliary mathematical tools which we will need in establishing the proof of Theorem 1 (see Section 3).
We need some properties of fractional calculus to provide a definition of global weak solution to problem (1) and (2).
Fixing , for given and , we recall the fractional integrals:
From Kilbas–Srivastava–Trjillo , for and , , one has
We recall that the inequality (15) holds for all and . On the other hand, by Lemma 1, we know that for all , the function defined by (10) belongs to . Moreover, we point out that
So, for all , the function (defined by (10)) belongs to . It follows that such a function is suitable as test function in (15), and (obviously) . Moreover, we take and greater than two threshold values, that is
Now, we combine such “ingredients” to obtain suitable estimates of the three terms in the right-hand side of (15).
Acting on the left-hand side of (21), it follows from (24) and (25) that
Hence, fixing and passing to the limit as in (26), we obtain
which contradicts the fact that . We conclude that problem (1) and (2) admits no global weak solution. □
The presented results confirm usefulness and simplicity of the test function method in analyzing different forms of wave equations. As a consequence of this approach in the previous sections, it is possible to see that the inhomogeneous damped wave equation with nonlinear memory (1) and (2) admits no global weak solution. In particular, we point out the absence of critical growth exponent for the nonlinear memory, in proving such a result.
This topic may be significant for the study of the controllability of solutions to certain nonlinear models of physics systems, together with the symmetry analysis. We have already mentioned in Section 1 above, a possible relationship between the boundary conditions, and the properties and behavior of a physical system. Here we imposed a Cauchy condition, but it will be interesting to know how Neumann, Robin and mixed boundary conditions affect the analysis of the wave type problem (1) and (2).
Investigation, all authors; Writing—original draft, all authors. All authors have read and agreed to the published version of the manuscript.
The first author is supported by Researchers Supporting Project number (RSP-2020/57), King Saud University, Riyadh, Saudi Arabia.
The authors are very grateful to the referees for the valuable comments and suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
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