scispace - formally typeset
Search or ask a question
Author

Ning Pan

Bio: Ning Pan is an academic researcher from Northeast Forestry University. The author has contributed to research in topics: Parabolic partial differential equation & p-Laplacian. The author has an hindex of 4, co-authored 4 publications receiving 96 citations.

Papers
More filters
Journal ArticleDOI
TL;DR: In this article, the existence of a global solution for a diffusion problem of Kirchhoff type driven by a nonlocal integro-differential operator was studied and the Galerkin method and potential well theory were used to obtain the solution.
Abstract: In this paper we study the existence of a global solution for a diffusion problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following parabolic equation involving the fractional p -Laplacian: { ∂ t u + [ u ] s , p ( λ − 1 ) p ( − Δ ) p s u = | u | q − 2 u , in Ω × R + , ∂ t u = ∂ u / ∂ t , u ( x , 0 ) = u 0 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where [ u ] s , p is the Gagliardo p –seminorm of u , Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω , p q N p / ( N − s p ) with 1 p N / s and s ∈ ( 0 , 1 ) , 1 ≤ λ N / ( N − s p ) , ( − Δ ) p s is the fractional p -Laplacian. Under some appropriate assumptions, we obtain the existence of a global solution for the problem above by the Galerkin method and potential well theory. It is worth pointing out that the main result covers the degenerate case, that is the coefficient of ( − Δ ) p s can vanish at zero.

44 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the wave problem of Kirchhoff type driven by a nonlocal integro-differential operator and obtained the global existence, vacuum isolating and blowup of solutions by combining the Galerkin method with potential wells theory.
Abstract: In this paper, we are concerned with a wave problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following hyperbolic problem involving the fractional Laplacian $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt} +[u]^{2 (\theta -1)}_{s}(-\Delta )^su=|u|^{p-1}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\quad u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N {\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array}\right. } \end{aligned}$$ where $$[u]_{s}$$ is the Gagliardo seminorm of u, $$s\in (0,1)$$ , $$\theta \in [1, 2_s^*/2)$$ , with $$2_s^*=2N/(N-2s)$$ , $$p\in (2\theta -1, 2_s^*-1]$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary $$\partial \Omega $$ , $$(-\Delta )^s$$ is the fractional Laplacian. Under some appropriate assumptions, we obtain the global existence, vacuum isolating and blowup of solutions for the above problem by combining the Galerkin method with potential wells theory. Finally, we investigate the existence of global solutions for the above problem with the critical initial conditions. The significant feature and difficulty of the above problem are that the coefficient of $$(-\Delta )^s$$ can vanish at zero.

32 citations

Journal Article
TL;DR: In this paper, the existence and uniqueness of weak solutions for nonlinear parabolic equations with p(x) growth in the space W 1,xLp(x)(Q) ∩ L∞(0, T ; L2(Ω)).
Abstract: In this article we study nonlinear parabolic equations with p(x)growth in the space W 1,xLp(x)(Q) ∩ L∞(0, T ; L2(Ω)). By using the method of parabolic regularization, we prove the existence and uniqueness of weak solutions for the equation ∂u ∂t = div(a(u)|∇u|p(x)−2∇u) + f(x, t). Also, we study the localization property of weak solutions for the above equation.

23 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the Kirchhoff-type wave problems with nonlinear damping and source terms involving the fractional Laplacian, and obtained the global existence, vacuum isolating, asymptotic behavior and blowup of solutions for the problem above by combining the Galerkin method with potential wells theory.
Abstract: In this paper, we consider the following Kirchhoff-type wave problems, with nonlinear damping and source terms involving the fractional Laplacian, $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{tt} +[u]^{2\gamma -2}_{s}(-\Delta )^su+|u_t|^{a-2}u_t+u=|u|^{b-2}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\ \ \ \ u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N{\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array} \right. \end{aligned}$$ where $$(-\Delta )^s$$ is the fractional Laplacian, $$[u]_{s}$$ is the Gagliardo semi-norm of u, $$s\in (0,1)$$ , $$2

23 citations

Journal ArticleDOI
TL;DR: In this article , the imbedding inequalities for homotopy operator are derived in Hölder-Morrey spaces on differential forms, and composite theorems which are associated with conjugate A -harmonic equations on differential form are given.
Abstract: Abstract The Hölder–Morrey spaces $\Lambda _{\kappa}^{p,\tau}(\Omega ,\wedge ^{l})$ Λ κ p , τ ( Ω , l ) are proposed in this paper. The imbedding inequalities for homotopy operator are derived in Hölder–Morrey spaces on differential forms. The Hölder continuity for Riesz potential with envelope function is deduced. As application, some composite theorems, which are associated with conjugate A -harmonic equations on differential forms, are given.

Cited by
More filters
Journal ArticleDOI
TL;DR: In this article, a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator is studied, and it is shown that the local nonnegative solutions blowup in finite time with arbitrary negative initial energy and suitable initial values.
Abstract: In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [u] s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.

98 citations

Journal ArticleDOI
TL;DR: In this article, the existence of a global solution for a diffusion problem of Kirchhoff type driven by a nonlocal integro-differential operator was studied and the Galerkin method and potential well theory were used to obtain the solution.
Abstract: In this paper we study the existence of a global solution for a diffusion problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following parabolic equation involving the fractional p -Laplacian: { ∂ t u + [ u ] s , p ( λ − 1 ) p ( − Δ ) p s u = | u | q − 2 u , in Ω × R + , ∂ t u = ∂ u / ∂ t , u ( x , 0 ) = u 0 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where [ u ] s , p is the Gagliardo p –seminorm of u , Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω , p q N p / ( N − s p ) with 1 p N / s and s ∈ ( 0 , 1 ) , 1 ≤ λ N / ( N − s p ) , ( − Δ ) p s is the fractional p -Laplacian. Under some appropriate assumptions, we obtain the existence of a global solution for the problem above by the Galerkin method and potential well theory. It is worth pointing out that the main result covers the degenerate case, that is the coefficient of ( − Δ ) p s can vanish at zero.

44 citations

Journal ArticleDOI
TL;DR: In this article, the existence and multiplicity results for the non-local p ( x ) -Kirchhoff problem were studied and an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.
Abstract: In this work, we study the existence and multiplicity results for the following nonlocal p ( x ) -Kirchhoff problem: (0.1) − a − b ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x d i v ( | ∇ u | p ( x ) − 2 ∇ u ) = λ | u | p ( x ) − 2 u + g ( x , u ) in Ω , u = 0 , on ∂ Ω , where a ≥ b > 0 are constants, Ω ⊂ R N is a bounded smooth domain, p ∈ C ( Ω ¯ ) with N > p ( x ) > 1 , λ is a real parameter and g is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.

36 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the wave problem of Kirchhoff type driven by a nonlocal integro-differential operator and obtained the global existence, vacuum isolating and blowup of solutions by combining the Galerkin method with potential wells theory.
Abstract: In this paper, we are concerned with a wave problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following hyperbolic problem involving the fractional Laplacian $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt} +[u]^{2 (\theta -1)}_{s}(-\Delta )^su=|u|^{p-1}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\quad u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N {\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array}\right. } \end{aligned}$$ where $$[u]_{s}$$ is the Gagliardo seminorm of u, $$s\in (0,1)$$ , $$\theta \in [1, 2_s^*/2)$$ , with $$2_s^*=2N/(N-2s)$$ , $$p\in (2\theta -1, 2_s^*-1]$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary $$\partial \Omega $$ , $$(-\Delta )^s$$ is the fractional Laplacian. Under some appropriate assumptions, we obtain the global existence, vacuum isolating and blowup of solutions for the above problem by combining the Galerkin method with potential wells theory. Finally, we investigate the existence of global solutions for the above problem with the critical initial conditions. The significant feature and difficulty of the above problem are that the coefficient of $$(-\Delta )^s$$ can vanish at zero.

32 citations