Abstract
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.
Highlights
We consider the following Cauchy problem for the fractional semilinear pseudo-parabolic equation:⎧ ⎨ut + k(– )aut + (– ⎩u|t=0 = u0,)au = up, x ∈ Rn, t > 0, (1.1)where p > 0, k > 0, u0(x) is sufficiently smooth and nonnegative
The pseudo-parabolic equation is used in diverse fields such as seepage theory of homogeneous liquid through cracked rock [3], the unidirectional propagation of nonlinear dispersive long waves [4, 5], and the description of racial migration [6]
We study the Green’s function of Cauchy problem (1.1) and obtain the following:
Summary
We consider the following Cauchy problem for the fractional semilinear pseudo-parabolic equation:. 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]. 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]. They proved the pointwise estimation of the solution of equation (1.1) at a. We make the following assumptions: (H1) u0 ∈ Cα+2(Rn) for sufficiently small u0 > 0; (H2) u0 ∈ W –s,2(Rn) ∩ W –s,∞(Rn) ∩ L∞(Rn) ∩ L2(Rn), 0 ≤ s < n, for sufficiently small u0 > 0.
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