Abstract
In this paper, we investigate a initial value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace operator of order 0 < ν ≤ 1 and the nonlinear memory source term. For 0 < ν < 1, the problem will be considered on a bounded domain of ℝd. By some Sobolev embeddings and the properties of the Mittag-Leffler function, we will give some results on the existence and the uniqueness of mild solution for problem (1.1) below. When ν = 1, we will introduce some Lp − Lq estimates, and based on them we derive the global existence of a mild solution in the whole space ℝd.
Highlights
In the last decades, the number of research works on derivatives of non-integer increased sharply
It has been shown that these new types of derivatives are more adequate than the classical integer ones
Fractional derivatives give us a potential instrument for the description of memory hereditary properties of various materials and processes, which are neglected with the classical integer derivative
Summary
The number of research works on derivatives of non-integer increased sharply. Let us list some works on the pseudo-parabolic equations with integer order. The boundary value problems for a third-order pseudo-parabolic equation with variable coefficients and with the Caputo fractional derivative were studied in [6]. INITIAL VALUE PROBLEM FOR FRACTIONAL VOLTERRA INTEGRODIFFERENTIAL PSEUDO-PARABOLIC EQUATIONS 3. For ν ∈ (0, 1), in [17], the authors introduced the time-weighted energy method to overcome the weakly dissipative property of the equation and established the global existence and time-decay rates for small-amplitude solutions to the Cauchy problem m of the fractional pseudo-parabolic equation on the whole space Rd, d ≥ 1. In the last section, when the domain D ≡ Rd, the global well-posed result will be given for problem (1.1) by using Lp − Lq estimates method
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