On mild solutions of the generalized nonlinear fractional pseudo-parabolic equation with a nonlocal condition

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

“…are oriented counterclockwise. From (15), it is easy to see that for t > 0 the integral in (17) converges in the uniform topology. Moreover,…”

“…. , l) are given constants, and 0 < τ 1 < τ 2 < • • • < τ l < T. In addition, for some applications of nonlocal conditions, please refer to [14][15][16][17][18].…”

On a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded delay

“…Many applications in practice can be embodied by the pseudo-parabolic equations, such as the aggregation of populations, 15 the unidirectional prapagation of nonlinear dispersive long waves, 16 and the incompressible fluids. 17 In addition, there are plenty of works focusing on the study of various pseudo-parabolic equations; one can refer to the papers [18][19][20][21][22][23][24][25][26][27][28][29] and the references therein. Especially, the Nguyen et al 30 investigated the well-posedness of the following time-fractional pseudo-parabolic equation with fractional Laplacian on a bounded domain Ω…”

“…Many applications in practice can be embodied by the pseudo‐parabolic equations, such as the aggregation of populations, 15 the unidirectional prapagation of nonlinear dispersive long waves, 16 and the incompressible fluids 17 . In addition, there are plenty of works focusing on the study of various pseudo‐parabolic equations; one can refer to the papers 18–29 and the references therein. Especially, the Nguyen et al 30 investigated the well‐posedness of the following time‐fractional pseudo‐parabolic equation with fractional Laplacian on a bounded domain $$$ \Omega $$$ $$$$ \left\{\begin{array}{ll}{}&amp;amp;amp;#x0005E;C{\partial}_{0,t}&amp;amp;amp;#x0005E;{\alpha}\left(u-m\Delta u\right)\left(x,t\right)&amp;amp;amp;#x0002B;{\left(-\Delta \right)}&amp;amp;amp;#x0005E;{\beta }u\left(x,t\right)&amp;amp;amp;#x0003D;\mathcal{N}(u),&amp;amp;amp; x\in \Omega, t&amp;amp;gt;0,\\ {}u\left(x,t\right)&amp;amp;amp;#x0003D;0,&amp;amp;amp; x\in \mathrm{\partial \Omega },t&amp;amp;gt;0,\\ {}u\left(x,0\right)&amp;amp;amp;#x0003D;{u}_0(x),&amp;amp;amp; x\in \Omega \end{array}\right.$$…”

Abstract fractional linear pseudo-parabolic equations in Banach spaces: well-posedness, regularity, and asymptotic behavior

“…In fact, in [14] and [27] a fractional Laplacian (−∆) α , α > 0, replaces the classical one acting both on u(x, t) and some functional of u(x, t) respectively, and the well-posedness and asymptotic behavior of its solutions is studied. In [6,8,11,24,31] the study is extended to semi-linear pseudo-parabolic equations also involving a fractional Laplacian. In [28,29] two different powers of the Laplacian acting separately on u(x, t) and ∂ t u(x, t) are considered, and in [18,23] time fractional derivatives are introduced in the format…”