On a final value problem for a nonlinear fractional pseudo-parabolic equation

In this paper, we investigate a final boundary value problem for a class of fractional with parameter \begin{document}$ \beta $\end{document} pseudo-parabolic partial differential equations with nonlinear reaction term. For \begin{document}$ 0 the solution is regularity-loss, we establish the well-posedness of solutions. In the case that \begin{document}$ \beta >1 $\end{document} , it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.

(Communicated by Runzhang Xu) Abstract. In this paper, we investigate a final boundary value problem for a class of fractional with parameter β pseudo-parabolic partial differential equations with nonlinear reaction term. For 0 < β < 1, the solution is regularityloss, we establish the well-posedness of solutions. In the case that β > 1, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.
1. Introduction. We consider the final value problem: in Ω, where m > 0, and Ω ⊂ R d , (d ≥ 1) is a bounded domain with smooth boundary ∂Ω, the operator (−∆) β with is the fractional Laplace operator with 0 < β = 1 and the final data ϕ ∈ L 2 (Ω). Pseudo-parabolic equations have many applications in science and technology, especially in physical phenomena such as seepage of homogeneous fluids through a fissured rock, aggregation of populations,... see e.g. Ting [24], R.
Xu [17,[31][32][33] and references therein. For β = 1, the direct problem is with conditions u(0, x) = u 0 (x), x ∈ Ω and u(t, x) = 0, (t, x) ∈ (0, T ] × ∂Ω. Problem (1.1) has been studied by many authors. Specifically, 1. f = 0, see e.g. [11,22,24], the existence and uniqueness of solutions is established. Moreover, the asymptotic behavior and regularity are investigated. 2. f (u) = u p , p ≥ 1, in [5], the authors investigate large time behavior of solutions. R. Xu et al. [32] proved the invariance of some sets, global existence, nonexistence and asymptotic behavior of solutions with initial energy J(u 0 ) ≤ d and finite time blow-up with high initial energy J(u 0 ) > d and some related works [18,34]. For the case of f (u) = |u| p−2 u, there are other results on the large time behavior of solutions of the pseudo-parabolic see [7, 28-30, 35, 36] and their references. 3. When the source term is a logarithmic nonlinearity f (u) = |u| p u log |u|, very recently, the work [10] focus on the initial conditions, which ensure the solutions to exist globally, blow up in finite time and blow up at infinite time. The asymptotic behaviour for the solutions has been considered in [4,6,8,12,34] and the references therein. 4. For nonlocal source, f (u) = |u| p Ω G(x, y)|u| p+1 (y)dy, y ∈ Ω, the authors of [19] considered blow-up time for solutions, obtained a lower bound as well as an upper bound for the blow-up time under different conditions, respectively. Also, they investigated a nonblow-up criterion and compute an exact exponential decay, see also [9,23].
For 0 < β = 1, [14] considered the Cauchy problem u t − m∆u t + (−∆) β u = u p+1 , in R + × Ω, (1.2) supplemented with initial condition u(0, x) = u 0 (x), x ∈ Ω and Dirichlet boundary condition u(t, x) = 0, (t, x) ∈ R + × ∂Ω. The paper established the global existence and time-decay rates for small-amplitude solutions. As mention above, initial value problems of nonlinear pseudo-parabolic equations have been considered in many papers see [1, 4-15, 19, 21-24, 32, 35-37]. However, there are not many results devoted to Problem (P T ). Our approach includes as special cases all previously on the reaction terms. In this work, we consider two cases; first, the source f is globally Lipschitz and in the second case, we consider f is general locally Lipschitz function (a coercive-type condition). At this point, we remark that there exists some locally Lipschitz functions, but we cannot determine its specific Lipschitz coefficient e.g. f = u(a − bu 2 ), b > 0 of the Ginzburg-Landau equation. Hence, we have to find another method to study the problem with the locally Lipschitz source which is similar to the Ginzburg-Landau equation, etc. (see Subsection 4.2.2 for more details).
The solution of Problem (P T ) is of the regularity-loss structure for 0 < β < 1, x ∈ Ω, t ∈ (0, T ], we consider the existence and regularity of Problem (P T ). In the case β > 1, the regularity-gain type for x ∈ Ω, t ∈ (0, T ] and the Problem (P T ) is ill-posed. So the regularization methods are required. As we know, there are many regularization methods to suit each problem [2,3,16,20,[25][26][27]. For problem (P T ), we propose two methods to regularize solution: Modified Lavrentiev regularization (MLR) method and Fourier truncated regularization (FTR) method.
The plan of the paper is as follows. Section 2 contains notations and formulation of a solution of Problem (P T ) and the proof of its instability. In Section 3, the case 0 < β < 1, well-posedness of solutions of Problem (P T ) is established. In Section 4, the case β > 1, the proof that Problem (P T ) is ill-posed and the well-posedness of the regularized problem are presented. We also propose two regularization methods: MLR method and FTR method for the globally Lipschitz or locally Lipschitz reaction terms, respectively.

Preliminaries.
2.1. Relevant notations. Let us recall that the spectral problem admits a family of eigenvalues The notation · B stands for the norm in the Banach space B. We denote by L q (0, T ; B), 1 ≤ q ≤ ∞, the Banach space of real-valued measurable functions w : The norm of the function space C k ([0, T ]; B), 0 ≤ k ≤ ∞ is denoted by For any ν ≥ 0, we define the space where ·, · L 2 (Ω) is the inner product in L 2 (Ω); H ν (Ω)) is a Hilbert space with the norm The Gevrey of order β class of functions with index η 1 , η 2 > 0, defined by the spectrum of the Laplacian is denoted by , and its norm given by Next, we give the formulation of solution of the problem (P T ).

2.2.
Mild solution of the Problem (P T ). Now, assume that Problem (P T ) has a unique solution, then we find its the form. Let u(t, x) = ∞ j=1 u j (t)e j (x) be the decomposition of u in L 2 (Ω) with u j (t) = u(t, ·), e j L 2 (Ω) . From (P T ), taking the inner product of both sides of (P T ) with e j (x), we obtain the ordinary differential equation where ϕ j = ϕ, e j L 2 (Ω) .
Definition 2.1 (Mild solution of Problem (P T )). A function u is a mild solution of (P T ) if u ∈ C([0, T ]; L 2 (Ω)) and satisfies the following integral equation for all (t, x) ∈ (0, T ) × Ω, and β > 0. Now we introduce the main results in this paper.
In this section, we prove that the Problem (P T ) is well-posed. First prove that for the Problem (P T ) exists a unique mild solution, then the regularity of the solution is established. We will make the following assumptions: (A 1 ) Assume that f satisfy the global Lipschitz condition: with K > 0 independent of t, x, w 1 , w 2 , and (t, Theorem 3.1 (L ∞ -Existence). Let 0 < β < 1, assume that f satisfy the assumption (A 1 ). Then, the integral equation (2.2) has a unique mild solution in L ∞ (0, T ; L 2 (Ω)).

3)
and we aim to show that H k is a contraction mapping on the space L ∞ (0, T ; L 2 (Ω)). In fact, we will prove that for every w 1 , w 2 ∈ L ∞ (0, T ; L 2 (Ω)), it holds

(3.4)
We will prove (3.4) by induction. For k = 1, using Parsevals relation and assumption (A 1 ), one obtains Assume now that (3.4) is satisfied for k = k 0 , let us prove that it is satisfied for k = k 0 + 1. It holds Therefore, by the induction principle we get (3.4). Since the right hand side of (3.5) is independent of t, we deduce that When k is large enough, we have 1 We claim that the mapping H k1 is a contraction, i.e. H k1 (u) = u. We have Due to the uniqueness of the fixed point of H k1 , it holds H (u) = u. We conclude that the integral equation (3.3) has a unique solution in L ∞ (0, T ; L 2 (Ω)).
4. The case β > 1: Ill-posedness and regularization methods for Problem (P T ).  iii) the solution's behaviour changes continuously with the initial conditions. If at least one of the three properties above does not satisfy, the problem is ill-posed.
It is easy to see that (here, noting that from (4.2), we have f j (0) = 0) Hence . This leads to We continue to estimate the right hand side of the latter inequality. Indeed, since {e j (x)} j≥1 is a basis of L 2 (Ω), i.e., e k , e j L 2 (Ω) = 1, k = j, e k , e j L 2 (Ω) = 0, k = j, we have Since the function χ(t) := exp (T − t)λ β k 1 + mλ k is a decreasing function with respect to variable t ∈ [0, T ] and β > 1, we deduce that Combining (4.4) and (4.5) yields As k ∞, we see that Thus, it is shown that Problem (P T )is ill-posed in the Hadamard sense in L 2 -norm for β > 1.

4.2.
Regularization and error estimate. In order to obtain the stable numerical solutions, we propose two regularization methods to solve the Problem (P T ) in two cases of f : where we set and the coefficient α := α(ε) satifies lim ε→0 + α = 0; it plays the role of a regularization parameter.
With the same argument as in the proof of (4.8), we obtain (4.9). This concludes the proof of the lemma.
Assume that (4.14) holds for k = N . We show now that (4.14) holds for k = N +1. In fact, we have By the induction principle, we deduce that (4.14) holds for all k ∈ N * . Notice that, as α is fixed, then tends to 0 when k ∞, so there exists a positive integer k 0 such that It means that F k0 is a contraction. Finally, it follows the desired conclusion that the problem (4.6) has a unique solution U ε α ∈ C([0, T ]; L 2 (Ω)). The proof is complete.
Since the right side of (4.21) does not depend on t, we have This leads to (4.16). b) From Lemma 4.2a), we first observe that The next observation, using Lemma 4.2b), is that (4.22) and (4.23), we deduce that . Thanks to Grönwall's inequality, we get which implies (4.17). This concludes the proof.
In order to estimate M α,ε 7 (t, x) and M α,ε 8 (t, x), we need to divide the proof in two steps.

Similar calculations yield
(4.31) Step 2. Estimate M α,ε 8 (t). Let us define an operator It clearly follows that The triangle inequality allows to write We estimate for M α 8a (t) as follows (similarly as in (4.30)): The term M α 8b (t) is estimated by using (4.8) as , for all j ∈ N * and η 1 ≥ 2, η 2 ≥ 2T mλ1 . Combining all these inequalities, we deduce Using Grönwall's inequality, we thus obtain We begin by establishing the locally Lipschitz properties of f by the following assumption: (A 3 ) Assume that for each > 0, there exists K > 0 such that and K is a non-decreasing function of . We assume that lim Example 1. Let f 1 (u) = u|u| 2 . Easy calculations show that Clearly, f 1 is not globally Lipschitz. For ≥ max{|u|, |v|}, from (4.33), one can compute explicitely the coefficient K = 3 2 .
Example 2. Let f 2 (u) = u(a − bu 2 ) (Ginzburg-Landau type), with a ∈ R, b > 0. It can be easily seen that f 2 is locally Lipschitz source. Moreover, we are possible to verify computationally the coefficient K = 3 2 max{|a|, b}. In order to solve the problem with the locally Lipschitz sources as above (and some other types), we outline our ideas to construct an approximation of the function f . For all > 0, we define if u ∈ ( , ∞). With this definition, we claim that f is global Lipschitz function. Before stating the main theorem, we first consider the following lemma.
Proof. The proof can be found in [2].
Remark 3. For ε > 0 satisfying lim Based on the above analysis, we propose the regularized solution by using FTR method as follows . The regularity estimates of the solution V ε Π α given by (4.36) is possible that but we will not develop this point here because it is an argument analogous to the previous one. We continue with the error estimate result.
Then the following stability estimate holds for any t ∈ [0, T ], δ > 0, Remark 5. Obviously, from (4.26), one can raise the question about I 0 in (4.25) is large, then the estimate in (4.26) is not exist naturally. By using FTR method, this problem is improved as in (4.38) by the term I0 (Λ ε ) β+δ → 0 as ε goes to 0 + . Proof. We divide the proof into two parts. In Part a, we prove that the integral equation (4.36) has a unique solution V ε Λ ε ∈ C([0, T ]; L 2 (Ω)). In Part b, the error between the exact solution and regularized solution is obtained. Part a. The existence and uniqueness of a solution of the integral equation (4.36).