Blow-up of solutions of nonlinear Sobolev type equations with cubic sources

Корпусов, М. О.; Sveshnikov, A. G.

doi:10.1134/s001226610603013x

Cited by 37 publications

(24 citation statements)

References 7 publications

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“…Eq. (1.1) can be used in the analysis of nonstationary processes in semiconductors in the presence of sources [15,16], where k ∂ u ∂t − ∂u ∂t corresponds to the free electron density rate, u corresponds to the linear dissipation of the free charge current and u p describes a source of free electron current. Furthermore, Eq.…”

Section: Introductionmentioning

confidence: 99%

Cauchy problems of semilinear pseudo-parabolic equations

Cao

Yin

Wang

2009

Journal of Differential Equations

126

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This paper concerns with the Cauchy problems of semilinear pseudo-parabolic equations. After establishing the necessary existence, uniqueness and comparison principle for mild solutions, which are also classical ones provided that the initial data are appropriately smooth, we investigate large time behavior of solutions. It is shown that there still exist the critical global existence exponent and the critical Fujita exponent for pseudo-parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations.

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Section: Introductionmentioning

confidence: 99%

Cauchy problems of semilinear pseudo-parabolic equations

Cao

Yin

Wang

2009

Journal of Differential Equations

126

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“…where k is a positive constant, Ω is a bounded domain of R n with smooth boundary ∂Ω and is the standard Laplace operator, can be used in the analysis of nonstationary processes in semiconductors in the presence of sources [21,22]. Furthermore, Eq.…”

Section: Introductionmentioning

confidence: 99%

Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

Chen¹,

Xu²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity ut − X ut − X u = u log |u|, where X = (X 1 , X 2 , • • • , Xm) is an infinitely degenerate system of vector fields, and X := m j=1 X 2 j is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincaré inequality, we obtain the global existence and blow-up at +∞ of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.

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“…Recently, there are some interesting results about the global existence and blow‐up of solutions for the problem with

f false(u false) = u^{p}

in where a family of potential wells is introduced to prove global existence, nonexistence and asymptotic behavior of solutions with low initial energy, while for high initial energy, finite time blow‐up of solutions is acquired by comparison principle. For other related works, we refer the readers to and the references therein. The obtained results show that global existence and nonexistence depend roughly on p , the degree of nonlinearity in f , the dimension n , and the size of the initial data.…”

Section: Introductionmentioning

confidence: 99%

Global existence, asymptotic stability and blow‐up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

Zhang

Liu

2019

Mathematische Nachrichten

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In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms.We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively. K E Y W O R D S blow-up, decay, generalized Boussinesq equation, global existence, nonlinear boundary condition M S C ( 2 0 1 0 ) 35K05, 35K61, 35K70

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Blow-up of solutions of nonlinear Sobolev type equations with cubic sources

Cited by 37 publications

References 7 publications

Cauchy problems of semilinear pseudo-parabolic equations

Cauchy problems of semilinear pseudo-parabolic equations

Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

Global existence, asymptotic stability and blow‐up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

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