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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 + ∫ 0 t g (t − τ) Δ B w (τ) d τ = | w | p − 1 w − 1 | B | ∫ B | w | p − 1 w 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 ∈ 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 ∈ 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. [ABSTRACT FROM AUTHOR]

Let d 2 {4; 5; 6...} and let X be a closed compact manifold of dimension d - 1. Let B = [0, 1) - = X be a conical domain and VB its interior. Consider the reaction-diffusion parabolic where T > 0, p 2 .1;1/, u0 belongs to the cone Sobolev space H 1;d=2 2;0. B/and ?B is the Fuchsian Laplacian. We investigate the existence of a global solution to (P) together with the decaying and blow-up properties using the potential well method. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with a class of Petrovsky system with nonlinear damping w t t + Δ B 2 w − k 2 Δ B w t + a w t | w t | m − 2 = b w | w | p − 2 $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$ on a manifold with conical singularity, where Δ B $\Delta _{\mathbb{B}}$ is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary x 1 = 0 $x_{1}=0$ . We first prove the global existence of solutions under conditions without relation between m and p, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy E ( 0 ) < d $E(0)< d$ .

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source wt−ΔBw−ΔBwt+∫0tg(t−τ)ΔBw(τ)dτ=|w|p−1w−1|B|∫B|w|p−1wdx1x1dx′ on a manifold with conical singularity, where the Fuchsian type Laplace operator ΔB is an asymmetry elliptic operator with conical degeneration on the boundary x1=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 w0∈W and Π(w0)>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 w0∈V and Π(w0)<0, the nonexistence of global solutions, i.e., the solutions blow up in finite time, is obtained by using the convexity technique.

This paper deals with the study of the initial-boundary value problem of edge-hyperbolic system with damping term on the manifold with edge singularity. More precisely, it is analyzed the invariance and vacuum isolating of the solution sets to the edge-hyperbolic systems on edge Sobolev spaces. Then, by using a family of modified potential wells and concavity methods, it is obtained existence and nonexistence results of global solutions with exponential decay and is shown the blow-up in finite time of solutions on the manifold with edge singularities. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of nonlinear viscoelastic evolution equation with strong damping and source terms, described by u tt - Δ B u + ∫ 0 t g (t - τ) Δ B u (τ) d τ + f (x) u t | u t | m - 2 = h (x) | u | p - 2 u , x ∈ int B , t > 0 , where B is a stretched manifold. First, we prove the solutions of problem (1.1) in the cone Sobolev space H 2 , 0 1 , n 2 (B) , which admit a blow up in finite time for p > m and positive initial energy. Then, we construct a lower bound for obtaining blow up time under appropriate assumptions on data. [ABSTRACT FROM AUTHOR]

In this paper, we will discuss about the invariance of solution set and present the existence and non-existence of the global solutions a class of initial-boundary value problems with dissipative terms is considered for a class of semilinear degenerate hyperbolic equations on the cone Sobolev spaces. First, we will discuss the invariance of some sets corresponding to the problem (1.1) and then, by using a family of potential wells and concavity methods, we obtain existence and non-existence results of global solutions with exponential decay and show the blow-up in finite time of solutions on a manifold with conical singularities. [ABSTRACT FROM AUTHOR]

In this paper, we apply the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations on the cone Sobolev spaces. We not only give some results of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions. Finally, we show blow-up in finite time of solutions on a manifold with conical singularities. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Let λ be the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator $ - \Delta _\mathbb{B} $ defined on a stretched cone $\mathbb{B}_0 \subseteq [0,1) \times X$ with boundary on { x = 0}. More precisely, $\Delta _\mathbb{B} = (x_1 \partial _{x_1 } )^2 \partial _{x_2 }^2 + \cdots + \partial _{x_n }^2 $ is also called the cone Laplacian. In this paper, by using Mellin-Fourier transform, we prove that $\lambda _k \geqslant C_n k^{\tfrac{2} {n}} $ for any k ⩾ 1, where $C_n = (\tfrac{n} {{n + 2}})(2\pi )^2 (|\mathbb{B}_0 |B_n )^{ - \tfrac{2} {n}} $, which gives the lower bounds of the Dirchlet eigenvalues of $ - \Delta _\mathbb{B} $. On the other hand, by using the Rayleigh-Ritz inequality, we deduce the upper bounds of λ, i.e., $\lambda _{k + 1} \leqslant (1 + \tfrac{4} {n})k^{2/n} \lambda _1 $. Combining the lower and upper bounds of λ, we can easily obtain the lower bound for the first Dirichlet eigenvalue $\lambda _1 \geqslant C_n (1 + \tfrac{4} {n})^{ - 1} 2^{\tfrac{n} {2}} $. [ABSTRACT FROM AUTHOR]

Abstract: We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples. [Copyright &y& Elsevier]