Well-posedness of the solution of the fractional semilinear pseudo-parabolic equation

This article concerns the Cauchy problem for the fractional semilinear pseudo-parabolic equation. Through the Green’s function method, we prove the pointwise convergence rate of the solution. Furthermore, using this precise pointwise structure, we introduce a Sobolev space condition with negative index on the initial data and give the nonlinear critical index for blowing up.

The pseudo-parabolic equation is used in diverse fields such as seepage theory of homogeneous liquid through cracked rock [3] (the coefficient of the third-order term represents the degree of cracks in the rock, and its decrease corresponds to the increase in the degree of cracking), the unidirectional propagation of nonlinear dispersive long waves [4,5] (where u is amplitude or curl), and the description of racial migration [6] (where u is the population density). Because of the wide range of applications of pseudo-parabolic equations, they attract great attention of mathematicians.
Ting, Showalter, and Gopala Rao proved the existence and uniqueness of the solution on the initial boundary value problem and the Cauchy problem of linear pseudo-parabolic equations, see [2,7,8]. Since then, many scholars have paid great attention to the study of nonlinear pseudo-parabolic equations, including about existence, asymptotic behavior, decay of regularity and solutions, etc., see [9][10][11][12]. Recently, Yang Cao et al. proved the existence and blowing up of the solution of equation (1.1) at a = 1, but the pointwise estimation of the solution was not discussed, see [13]. Later, Wang Weike et al. used the Green's function to improve [13] and p only needs to satisfy p > 1 + 2 n+s instead of p > 1 + 2 n . More specifically, they proved the pointwise estimation of the solution of equation (1.1) at a = 1, also obtained the nonlinear critical index of the blowing up at p > 1 + 2 n+s by limiting the initial condition, see [14].
In the above research, they focused on integer order equations. The fractional dissipation operator (-) a can be regarded as the infinitesimal generators of Levy stable diffusion process. Compared with the integral differential equation, it can describe some physical phenomena more accurately, see [15][16][17]. Therefore, more and more scientists are devoted to the research of fractional differential equations, see [16][17][18].
Motivated by the above works, we study the pointwise estimate and exponential decay of the solution for problem (1.1) in the fractional order case. At present, there is little research on the pointwise estimate and exponential decay of the solution of this fractional equation, and the main difficulty stems from its fractional dissipation operator term. The structure of this article is organized as follows: In Sect. 2, we recall some preliminary results and show the main results of this paper. In Sect. 3, by Green's function method, we use the Green's function to express the solution of fractional equation (1.1) and get the pointwise estimate result of the Green's function. In Sect. 4, we obtain the pointwise estimate of fractional equation (1.1) with appropriate conditions p, u 0 . In Sect. 5, we prove that the exponential decay of equation (1.1) still exists without a = 1.

Preliminaries and main results
Let C represent a generic positive constant, which may change from line to line. The norm of L p (Ω) is written as · L p (Ω) (1 ≤ p ≤ ∞). The notation X is a Banach space with a norm · X .
Then the Fourier transform is as follows: its inverse Fourier transform is According to [19], we have the following two lemmas.

4)
where k and m are any positive integers, (a) + = max(0, a), and then there exist distributions f 1 (x) and f 2 (x) satisfying where C 0 is a constant and δ(x) is the Dirac function. Furthermore, choosing ε 0 small enough, we have the estimate for positive integer 2N > n + |α|.
where N > 0 is an arbitrary constant, C depends on the initial value u 0 and the parameter p.

8)
where C depends on the initial value data and p.

Pointwise estimate of the Green's function
In this section, we will consider the pointwise estimation of the solution to linear form of problem (1.1). We study the Green's function of Cauchy problem (1.1) and obtain the following: is the Dirac function and ⊗ represents the tensor product. Considering the Fourier transform of equation (3.1) with respect to x, we get By solving the above equation directly, we know that where μ(ξ ) = -|ξ | 2a 1+k|ξ | 2a . Now we use frequency decomposition to obtain an estimate of the Green's function G. Let where χ 1 (ξ ) and χ 3 (ξ ) are the smooth cut-off functions, ε, R are any positive constants sat- From the literature [20], we know that the attenuation of the solution of the linear problem is mainly related to the low frequency part of G(t, ξ ). We use cut-off functions to divide the solution into three parts: low frequency, intermediate frequency, and high frequency.

Proposition 3.1 Let ε be a sufficiently small constant. Then there exists a constant C > 0 satisfying
Proof In the case of low frequency, let 0 < |ξ | < 2R, then G has compact support. Taking into account (3.3) and Lemma 2.1, there is for ∀|β| ≤ 2N . Then is (3.7) established.
Actually, we can discuss obtaining (3.7) in two cases.

15)
where m 0 is a positive constant.
Proof Choosing m sufficiently large and m > 1 2m . This analysis reflects that Now, we apply mathematical induction to prove the following inequality: Obviously, the above formula holds when |β| = 0. Suppose that |β| ≤ l -1, the above formula still holds. Then we will prove that the formula of (3.18) also holds for |β| ≤ l.
Now considering the high frequency part G 3 (x, t).

25)
where b is a positive constant and Expanded in Taylor's series at |ρ| → 0, we infer where p j (t) is a polynomial of degree j. Let With the help of Lemma 2.2 and choosing R big enough, it is easy to see that In conclusion, we use the following lemma to explain the estimate of the regular part of G.

32)
where F l is the distribution and Using Hausdorff-Young's inequality, the following lemma of Green's function is easily obtained.

Pointwise estimation of the solution
In this section, we get the pointwise estimate of the solution under appropriate conditions of u 0 , p.
On the other hand, It follows from (4.3) and (4.4) that From t 4 -M 2 ≥ 0, the proof is obtained.
Proof of Theorem 1 With the help of Lemma 3.3 and considering (1.1), from Green's function, we have where the symbol * represents convolution, F(u) = u p , and H satisfies Applying the inverse Fourier transform, we deduce Obviously, the estimated value on G is also correct on H. By using D α x to equation (4.8), we get (4.12) Then x H(xy, ts)φ p (y, s) dy ds. First of all, we discuss the singular part. By (H2), following [20], we have (4.14) Secondly, we consider the nonsingular part. Since |u 0 | ≤ (1 + |y| 2a ) -N , supp u 0 ⊂ {|y| ≤ M}, according to the definition of tight support, we can know u 0 has a compact support. If t is large enough, we find Here we still divide the nonlinear term into a singular part and a nonsingular part. Define where ψ 1,1 = t 0 R n D α x (G -F l )φ p (y, s) dy ds, ψ 1,2 = t 0 R n D α x F l φ p (y, s) dy ds. Estimating the nonsingular part. Recalling Lemma 3.1, it shows that We will discuss it in two cases: (1) If |x| 2a ≤ 1 + t, then (2) If |x| 2a > 1 + t, then (4.20) From (4.19)-(4.20), we can get Due to p > 1 + 2a n , we deduce Estimating the singular part. By [19], we know that Coming back to the whole solution, we find Therefore, Since M(0) ≤ Cε, applying the continuity method, we get that By the above inequality, we have (4.27)

Improvement of the initial data
In this section we consider the Cauchy problem of (1.1). It shows that the limit of the parameter p can be weaker when the initial conditions become stronger. Since we have known the proof of the existence and uniqueness of the solution, we will not discuss it.
Only for attenuation of the decay estimate.