Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

Alimohammady, Mohsen; Jafari, Ali; Kalleji, Morteza Koozehgar

doi:10.1007/s13226-017-0215-x

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…Another interesting type of model is the evolution equation with conical singularity (see [17][18][19][20][21][22][23][24][25][26][27]). Chen et al established some classic inequalities on the cone Sobolev spaces in [17,18].…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

2023

Symmetry

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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 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.

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Section: Introductionmentioning

confidence: 99%

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

Zhang

2023

Symmetry

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“…They discussed the invariance of some sets, global existence, nonexistence, and asymptotic behavior of solutions with initial energy J(w 0 ) < d by introducing a family of potential wells which was first proposed by Sattinger [24]. More works on equations with conical degeneration can be seen in the literature [25][26][27][28] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration

Shang

2020

Bound Value Probl

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This paper deals with a class of Petrovsky system with nonlinear damping w tt + 2 B w-k 2 B w t + aw t |w t | m-2 = bw|w| p-2 on a manifold with conical singularity, where B is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary 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.

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“…In [2], authors studied the problem 1.1 without damping term and the particular case of nonlinear term f (x, u) = g(x)|u| p−1 u. Furthermore,in the case of manifolds with conical singularities B, the well-known operator ∆ B + V (x) appears naturally in the nonlinear heat and wave equations, nonlinear and nonhomogeneous Schrödinger equations. For example, investigations have been done about the existence results, multiple solutions for nonhomogeneous degenerate Schrödinger equations in noncritical and critical cone Sobolev exponent on manifolds with conical singularities [3,8,15]. For finding such positive potential function any one can consider Poincaré's constant on manifold B [2].…”

Section: Introductionmentioning

confidence: 99%

Vacuum isolating and blow-up analysis for edge hyperbolic system on edge Sobolev spaces

Kalleji¹,

Kadkhoda²

2020

Turk J Math

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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.

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Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

Cited by 4 publications

References 10 publications

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

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

On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration

Vacuum isolating and blow-up analysis for edge hyperbolic system on edge Sobolev spaces

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