GLOBAL SOLUTIONS FOR FRACTIONAL VISCOELASTIC EQUATIONS WITH LOGARITHMIC NONLINEARITIES

. In this article we study a fractional viscoelastic equation of Kirch-hoﬀ type with logarithmic nonlinearities. Under suitable conditions we prove the existence of global solutions and the exponential decay of the energy.

The questions of solvability and the long time behavior of solutions to the doubly nonlinear nonlocal parabolic equation (ϕ(u)) t − div σ(∇u) = t 0 g(t − τ )divσ(∇u(τ ))dτ + f (x, t, u), were studied in [4,19,30,31,32,34].This equation arises from the study of heat conduction in materials with memory.On the other hand, many fractional and nonlocal operators are actively studied in recent years.This type of operators arises in a quite natural way in many interesting applications, such as, finance, physics, game theory, Lévy stable diffusion processes, crystal dislocation; see [5,22,36] and their references.The first result concerning fractional Kirchhoff problems was obtained by Fiscella and Valdinoci [18].Pan et al [26] investigated for the first time the existence of global weak solutions for degenerate Kirchhoff-type diffusion problems involving fractional p-Laplacian, by combining the Galerkin method with potential well theory, for the special function M (t) = t λ−1 (t ≥ 0).Mingqi et al [24] proved the existence and blow-up of solutions for a similar equation with more general conditions on M which cover the degenerate case.Recently, logarithmic nonlinearity appears frequently in partial differential equations which describes important physical phenomena, see [12,14,20,23,37] and the references therein.Ding and Zhou [14] studied the semilinear parabolic problem of Kirchhoff type with logarithmic nonlinearity, They obtained results of global solutions and of finite time blow-up of solutions, when the initial energy is subcritical and critical, by using the potential well method.In the works mentioned above, there are only a few about global existence and exponential decay rate for doubly nonlinear parabolic equations involving variable exponent, viscoelastic term in the fractional setting, and logarithmic nonlinear terms.Motivated by this, we study global solutions for (1.1) by using Galerkin's method and similar arguments as those in Tartar [33].Also, we give the exponential decay rate of the energy via the energy perturbation method.It is worth mentioning that we do not use the logarithmic Sobolev inequality to obtain our results.
The article is organized as follows.In Section 2, we give the preliminaries for our research.In Section 3, by using the Galerkin approximation method we obtain a global solution, and finally, we obtain the exponential decay under certain class of initial data.

Preliminaries
In this section, we present some material and assumptions needed in the rest of this paper.We denote Q = R 2N \ (CΩ × CΩ), CΩ := R N \ Ω, and where u| Ω represents the restriction to Ω of function u(x).Also, we define the linear subspace of W , W 0 = u ∈ W : u = 0 a.e. in R N \ Ω .The linear space W is endowed with the norm It is easily seen that • W is a norm on W and C ∞ 0 (Ω) ⊆ W 0 .The functional , is a equivalent norm on W 0 = {u ∈ W : u(x) = 0 a.e. in R N \ Ω} which is a closed linear subspace of W . Furthermore (W 0 , • W0 ) is a Hilbert space with inner product dxdy.
Now we review the main embedding results for the space W 0 .
Lemma 2.1 ([27, 28, 29]).The embedding and compact for any r . Now, we recall some background concerning the generalized Lebesgue-Sobolev spaces.We refer the reader to [15,16,17] for details.Set and the space which is a Banach space [21].We also define the space We denote by W , where (Ω) and the embedding is continuous.
The above lemma immediately follows from Lemma 2.2 and Young's inequality.
Then, for any s, τ ∈ [0, T ] with s < τ we have formula of integration by parts, |w(s)| r(x) dx.

Existence of global solutions and exponential decay
In this section, we focus our attention on global solutions and exponential decay for problem (1.1).
s [ , then problem (1.1) has a unique weak solution u for T small enough.
Proof.We prove the existence of weak solutions by using the Faedo-Galerkin method with ideas from [7].We choose a sequence and {w ν } is a standard orthonormal basis with respect to the Hilbert space L 2 (Ω) and an orthogonal basis in W 0 , where V m = spam{w 1 , w 2 , ....w m }.Now, we construct approximate solutions u m (m = 1, 2, . . .), of the problem (1.1), in the form where the coefficient functions g jm satisfy the system of ordinary differential equations Let us show that the system (3.1) is locally solvable.It is clear that (3.1) can be rewritten in the form where This system is equivalent to and the fact that the map s → f (s) is increasing for large s, we obtain So, by the elementary inequality s log s ≥ s − 1, ∀s > 0, we deduce that Φ is monotone coercive.Also it is obviously continuous.So, by the Brouwer theorem Φ is onto.In view of (3.3), Φ −1 is locally Lipchitz continuous.
Consider the map L : C(0, T, R m ) → C(0, T, R m ), defined by It is not hard to prove that L is completely continuous and that there exist (sufficient small) T m > 0 and (sufficient large) R > 0 such that L(B R ) ⊆ B R , where B R is the ball in C(0, T m , R m ) with center the origin and radius R. Consequently, by Schauder's theorem, the operator L has a fixed point in C(0, T m , R m ).This fixed point is a solution of (3.2).So, we can obtain an approximate solution u m (t) of (3.1) in V m over [0, T m [ and it can be extended to the whole interval [0, T ], for all T > 0, as a consequence of the a priori estimates that shall be proven in the next step.
First estimate.Multiplying (3.1) by g jm (t) and adding in j = 1, . . ., m, we have which implies, integrating with respect to the time variable from 0 to t on both sides, using Lemma 2.7 that where Let us introduce the function Θ(λ) = λ 0 g(λ−τ ) u m (τ ) W0 .Estimating the second term on right-hand side of (3.5) we have But, using Young Inequality and noting that ∞ 0 g(τ )dτ < 1, we obtain (3.9) To estimate the last term in (3.9) we use Lemma 2.6, where Taking suitably small in (3.10), it follows from (3. where Integrating (3.13) on [0, t], t ≤ T 0 we obtain From the assumptions on u 0 , (3.8), Lemma 2.6 and the estimate (3.12), it follows that for some constant M 1 > 0. By the above estimates (3.12) and (3.14), {u m } have subsequences still denoted by {u m } such that (3.15) Also, reasoning as in [12], taking into account the compact embedding of W 0 into L β (Ω), β = ρ, σ, we have Employing the same arguments as in [9] we can prove that Therefore, passing to the limit in (3.1) as m → +∞, by (3.15)-(3.17),we can show that u satisfies the initial condition u(0) = u 0 and The uniqueness property of solutions can be derived as in [13, Theorem 3, p. 1095], observing that u w0 )(−∆) s u is a monotone operator.We omit the details.
Next, we consider the existence of global solutions and their energy decay for problem (1.1).For this purpose we define the energy associated with problem (1.1) by Then, we easily can check that for any regular solution.This remains valid for weak solutions by simple density argument.This shows that E(t) is a nonincreasing function.
Before going on, we introduce the following notation Define the function we obtain that these λs satisfy the inequality and h (λ) ≥ 0 for 0 ≤ λ ≤ λ 1 where 1) .
Theorem 3.3.Assume that hypotheses of Theorem 3.2 are satisfied.Consider u 0 ∈ W 0 , satisfying where (3.26) Then the problem admits a global weak solution in time.In addition, if there exists a constant ξ 0 > 0 such that g (t) ≤ −ξ 0 g(t), then this solution satisfies where L 0 and γ are positive constants.
Proof.We will get global estimates for u m (t) solution of the approximate system (3.1)under the conditions (3.24)-(3.25)for u 0 .For this, it suffices to show that where E m (t) is defined in (3.18) with u(t) replaced by u m (t), is bounded and independently of t.From (3.13) and the definition of energy, we have From the convergence u 0m → u 0 in W 0 we see that E m (0) < l 4 λ 2 1 for sufficiently large m.We claim that there exists an integer ν 0 such that where C is a constant independent of m.Thus, we obtain the global solution.
Hence, by continuity there exists By (3.23), we see that . Therefore our claim is true.The above estimates permit us to pass to the limit in the approximate equation.
To show the uniform decay of the solution we introduce the perturbed energy functional where is a positive constant which shall be determined later, and