MDPI - Publisher of Open Access Journals

19 pages, 340 KB

Open AccessArticle

Nonlocal Pseudo-Parabolic Equation with Memory Term and Conical Singularity: Global Existence and Blowup

by Jiali Yu and Jihong Zhang

Symmetry 2023, 15(1), 122; https://doi.org/10.3390/sym15010122 - 1 Jan 2023

Cited by 3 | Viewed by 2206

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source [...] Read more.

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source

w_{t} - Δ_{B} w - Δ_{B} w_{t} + \int_{0}^{t} g (t - τ) Δ_{B} w (τ) d τ = {| w |}^{p - 1} w - \frac{1}{| B |} \int_{B} {| w |}^{p - 1} w \frac{d x_{1}}{x_{1}} d x^{'}

on a manifold with conical singularity, where the Fuchsian type Laplace operator

Δ_{B}

is an asymmetry elliptic operator with conical degeneration on the boundary

x_{1} = 0

. Firstly, we discuss the symmetrical structure of invariant sets with the help of potential well theory. Then, the problem can be decomposed into two symmetric cases: if

w_{0} \in W

and

Π (w_{0}) > 0

, the global existence for the weak solutions will be discussed by a series of energy estimates under some appropriate assumptions on the relaxation function, initial data and the symmetric structure of invariant sets. On the contrary, if

w_{0} \in V

and

Π (w_{0}) < 0

, the nonexistence of global solutions, i.e., the solutions blow up in finite time, is obtained by using the convexity technique. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

Search Results (1)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (1)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI