Runzhang Xu

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Verified

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• PhD
• Professor (Full) at Harbin Engineering University

In this paper, we study a class of semilinear pseudo-parabolic equations. By introducing a family of potential wells, we prove the invariance of some sets, global existence, nonexistence and asymptotic behavior of solutions with initial energy J(u0)⩽dJ(u0)⩽d. Moreover, we obtain finite time blow-up with high initial energy J(u0)>dJ(u0)>d by compari...

We consider the initial boundary value problem of pseudo-parabolic equation with singular potential. We obtain global existence, asymptotic behavior and blowup of solutions with initial energy J(u0)≤d. Moreover, we estimate the upper bound of the blowup time for J(u0)<0 and 0<J(u0)<d respectively. Finally, we prove the finite time blowup and estima...

We consider weak solutions of the Novikov equation that lie in the energy space \(H^1\) with non-negative momentum densities. We prove that a special family of such weak solutions, namely the peakons, is \(H^1\)-asymptotically stable. Such a result is based on a rigidity property of the Novikov solutions which are \(H^1\)-localized and the correspo...

This paper is concerned with a class of semilinear hyperbolic equations with singular potentials on the manifolds with conical singularities, which was introduced to describe a field propagating on the spacetime of a true string. We prove the local existence and uniqueness of the solution by using the contraction mapping principle. In the spirit of...

The b-family-Kadomtsev–Petviashvili equation (b-KP) is a two dimensional generalization of the b-family equation. In this paper, we study the spectral stability of the one-dimensional small-amplitude periodic traveling waves with respect to two-dimensional perturbations which are either co-periodic in the direction of propagation, or nonperiodic (l...

In this paper, we consider the initial boundary value problem for the 2-D hyperbolic viscous Cahn-Hilliard equation. Firstly, we prove the existence and uniqueness of the local solution by the Galerkin method and contraction mapping principle. Then, using the potential well theory, we study the global well-posedness of the solution with initial dat...

In the present paper, we study the well-posedness of the solution to the initial boundary value problem for the damped Kirchhoff-type wave equation with fractional Laplacian. First, the existence and uniqueness of the local solution are established by the Banach fixed point theorem. Then, the global existence and finite time blowup of the solution...

In this paper, we conduct a comprehensive study of the global well‐posedness of solution for a class of nonlocal wave equations with variable‐order fractional Laplacian and variable exponent nonlinearity by constructing a suitable framework of the variational theory. We first prove the local‐in‐time existence of the weak solution via the Galerkin a...

We are concerned with the description of global quantitative stability of wave equations with linear strong damping and linear or nonlinear weak damping. By giving some energy decay estimates, we obtain several conclusions about the continuous dependence of the global solution on the initial data and the coefficients of the strong damping term and...

This paper focuses on a class of generalized nonlinear wave equations with doubly dispersive over equation whole lines. By employing the potential well theory, we classify the initial profile such that the solution blows up or globally exists.

The viscous dissipation limit of weak solutions is considered for the Navier-Stokes equations of compressible isentropic flows confined in a bounded domain. We establish a Kato-type criterion for the validity of the inviscid limit for the weak solutions of the Navier-Stokes equations in a function space with the regularity index close to Onsager’s...

This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equations with rotational inertia, memory, p-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and the contraction mapping principle...

In this paper, we consider an initial boundary value problem of m-Laplacian parabolic equation arising in various physical models. We tackle this problem at three different initial energy levels. For the sub-critical initial energy, we obtain the blow-up result and estimate the lower and upper bounds of the blow-up time. For the critical initial en...

This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equation with rotational inertia, memory, $p$-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and contraction mapping principle. T...

In this paper, we establish two conclusions about the continuous dependence on the initial data of the global solution to the initial boundary value problem of a porous elastic system for the linear damping case and the nonlinear damping case, respectively, which reflect the decay property of the dissipative system. Additionally, we estimate the lo...

We propose a new differential inequality that improve the upper equations with strain term and arbitrary positive initial energy. We also give two new initial conditions to expand the range of the initial data leading to the �finite time blowup of solutions. We obtain a sharp result of fi�nite time blowup for the special case of the new differentia...

For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invarian...

This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, w...

Gas hydrate is an abundant natural resource that has attracted much attention around the world. This paper aims at analyzing the gas production potential of methane hydrate reservoir by depressurization. The physical model of a cylindrical reactor is established and numerically simulated by using TOUGH + HYDRATE_v1.5. Based on the experimental para...

It is important to study the temperature stratification phenomenon in flash evaporation to well design the marine accumulator and some related industrial equipment. Therefore, a series of flash evaporation experiments with initial temperature of 65.0 °C ~ 84.4 °C and superheat degree of 5.0 °C ~ 30.0 °C are carried out to study the temperature vari...

Studied herein is the local well-posedness of solutions with the low regularity in the periodic setting for a class of one-dimensional generalized Rotation-Camassa-Holm equation, which is considered as an asymptotic model to describe the propagation of shallow-water waves in the equatorial region with the weak Coriolis effect due to the Earth's rot...

Natural gas hydrate as a potential backup energy has very important scientific and social values. This paper aims at analyzing the gas production characteristic of methane hydrate sediment via depressurization. The physical model of 1 L cylindrical reactor is established and numerically simulated with the hydrate calculation software TOUGH+HYDRATE_...

We study the semilinear strongly damped plate equation by considering its two different problems. For initial value problem, we prove the local well-posedness and blow-up results of solution for the problem with polynomial nonlinear source terms. For terminal value problem, given the ill-posedness in the sense of Hadamard we propose a regularizatio...

In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the...

We discuss global and blowup solutions for general Gierer–Meinhardt system with zero Neumann boundary conditions, which originally arose in studies of pattern-formation in biology and has interesting and challenging mathematical properties. We obtain some new sufficient conditions for global existence and finite time blow-up of solutions. In some s...

This paper pays attention to the Cauchy problem for a class of nonlinear wave equations with linear pseudo-differential operator. In the framework of variational arguments, the existence and nonexistence of global solution to this problem are derived when the total initial energy is less than or equal to the mountain pass level. Further, a finite-t...

This paper investigates the initial boundary value problem for a class of fourth order nonlinear damped wave equations modeling longitudinal motion of an elasto-plastic bar. By applying a suitable potential well-convexity method, we derive the global existence, asymptotic behavior and finite time blow up for the considered problem with more general...

To understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity, we aim to study the global well-posedness of nonlinear fourth order wave equations with logarithmic source term, where the dispersive, the nonlinear weak damping and linear strong damping are taken into account. Based on the potent...

In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on...

For the nonlinear Kirchhoff-type wave system with logarithmic nonlinearities and weak dissipation the global well-posedness of initial boundary value problem is analyzed. Focusing on the interplay between Kirchhoff terms and logarithmic sources, we investigate the Kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in...

We study the Cauchy problem of nonlinear Schrödinger equation \(i\varphi _t+\Delta \varphi +|\varphi |^{p-1}\varphi =0\). By constructing infinite Nehari manifolds with geometric features, we not only obtain infinite invariant sets of solutions, but also give infinite sharp conditions for global existence and finite time blow up of solutions.

In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get th...

In this paper, we consider the following Kirchhoff-type wave problems, with nonlinear damping and source terms involving the fractional Laplacian, utt+[u]s2γ-2(-Δ)su+|ut|a-2ut+u=|u|b-2u,inΩ×R+,u(·,0)=u0,ut(·,0)=u1,inΩ,u=0,in(RN\Ω)×R0+,where (-Δ)s is the fractional Laplacian, [u]s is the Gagliardo semi-norm of u, s∈(0,1), 2<a<2γ<b<2s∗=2N/(N-2s), Ω⊂R...

The combustion of double fuel droplets with different initial centre spacings and different droplet radii in convective environment are studied numerically by using the VOF (volume of fluid) model. The influences of these two factors on the droplet burning rates in the process of spray combustion are studied. The combustion processes of the two dro...

This paper deals with blow-up of positive solutions for a singular Gierer–Meinhardt system subject to zero Dirichlet boundary conditions. We first prove the existence of a local solution and then show blow-up solutions under certain conditions for parameters. We use a functional method to obtain a local solution which is bounded by the first eigenf...

We consider the local well-posedness of the one-dimensional non-isentropic compressible Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic co...

The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E (0) < d , critical initial energy E (0) = d and the arbitrary high initial energy E (0) > 0 ( ω = 0). Firstly, we prov...

By introducing a new increasing auxiliary function and employing the adapted concavity method, this paper presents a finite time blow up result of the solution for the initial boundary value problem of a class of nonlinear wave equations with both strongly and weakly damped terms at supercritical initial energy level. © 2018 American Institute of M...

In this paper we consider the semilinear wave equation with logarithmic nonlinearity. By modifying and using potential well combined with logarithmic Sobolev inequality, we derive the global existence and infinite time blow up of the solution at low energy level E(0) < d. Then these results are extended in parallel to the critical case E(0) = d. Be...

The initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) with nonlinear coupled source terms is considered in order to classify the initial data for the global existence, finite time blowup and long time decay of the solution. The whole study is conducted by considering three cases according to initial...

Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and f...

By revealing that a key parameter D in Chen-Guo model is related to the temperature T, a new temperature-dependent parameter D(T) trained by the reported experimental data is proposed with Chen-Guo model to improve the prediction accuracy of phase equilibrium conditions of methane hydrate in the range of 145.75 K- 272.2 K. Four equations of state (...

We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrt{\rho} \in L^\infty_t H^1_x$. The main idea is to constru...

We consider the local well-posedness of the one-dimensional nonisentropic Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic conservation law...

In this paper, we study the initial boundary value problem and global well-posedness for a class of fourth order wave equations with a nonlinear damping term and a nonlinear source term, which was introduced to describe the dynamics of a suspension bridge. The global existence, decay estimate, and blow-up of solution at both subcritical (E(0) < d)...

The main goal of this work is to investigate the initial boundary value problem of fourth order wave equation with nonlinear strain and logarithmic nonlinearity at three different initial energy levels, i.e., subcritical energy E(0)<d, critical initial energy E(0)=d and arbitrary high energy E(0)>d. First, we prove the local existence of weak solut...

We establish a new finite time blowup theorem for the solution to the initial boundary value problem of a class of semilinear pseudo-parabolic equations at high initial energy level. And we also estimate the upper bound of the blowup time.

This article studies the existence and nonexistence of global solutions to the initial boundary value problems for semilinear wave and heat equation, and for Cauchy problem of nonlinear Schrodinger equation. This is done under three possible initial energy levels, except the NLS as it does not have comparison principle. The most important feature i...

In this paper, we study blow up and blow up time of solutions for initial boundary value problem of Kirchhoff-type wave equations involving the fractional Laplacian (equation presented), where [u]s is the Gagliardo seminorm of u, s ∈ (0, 1), θ ∈ [1, 2∗ s/2) with 2∗ s = N ² − N 2s, (−∆)s is the fractional Laplacian operator, f(u) is a differential f...

This paper is concerned with the finite time blow up of the solution to the Cauchy problem for the Klein–Gordon equation at arbitrarily positive initial energy level. By introducing a new auxiliary function and an adapted concavity method we establish some sufficient conditions on initial data such that the solution blows up in finite time, which e...

This paper considers the initial boundary value problem of solutions for a class of sixth order 1-D nonlinear wave equations. We discuss the probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and non-global existence of solutions at three different initial energy levels, i.e., sub-...

In this paper, we are concerned with the singular parabolic system ut=δu+f(x)v-p,vt=δv+g(x)u-q in a smooth bounded domain Ω⊂RN subject to zero Dirichlet conditions, with initial conditions u0(x),v0(x)>0. This problem is of interest as it is related to some problems in biology and physics. Under suitable assumptions on p,q and f(x),g(x), some existe...

We study the Cauchy problem of damped generalized Boussinesq equation utt − uxx + (uxx + f(u))xx − αuxxt = 0. First we give the local existence of weak solution and smooth solution. Then by using potential well method and convexity method we prove the global existence and finite time blow up of solution, then we obtain some sharp conditions for the...

This paper proves the global existence of solution for a class of nonlinear wave equations with nonlinear combined power-type nonlinearities of different signs for the initial data at sup-critical energy level.

In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applying the comparison principle, the maximum principle and the supersolution-subsolution method, we prove the global existence and blow u...

This paper is concerned with existence results for a singular Gierer–Meinhardt system subject to zero Dirichlet boundary conditions, which originally arose in studies of pattern-formation in biology. The mathematical difficulties are that the system becomes singular near the boundary and it lacks a variational structure. We use a functional method...

In this paper, a mathematical model of Rayleigh wave propagation in a sandy earth crust has been investigated under the influence of quadratically varying rigidity and linearly varying density in the incompressible half space. The upper boundary plane of the sandy crust has been assumed to be rigid. The displacement of the wave in two medium have b...

The present paper presents a study of Rayleigh wave propagation in a sandy crustal layer of the Earth lying over an orthotropic mantle with irregular boundary surfaces under the influence of initial stress, gravity field and rigid boundary plane. An attempt has been made to study the dynamics of the individual medium and derive the displacement of...

This paper studies dispersion of a G-type earthquake wave under the influence of a suppressed rigid boundary. Inside the Earth, the density and rigidity of the crustal layer and the mantle of the Earth vary exponentially and periodically along the depth. The displacements of the wave are found in the individual medium followed by a dispersion equat...

In this remark, we correct the proof of the asymptotic behavior of solution for the initial boundary value problem of semilinear pseudo-parabolic equations with critical initial energy obtained in Xu and Su (2013) [1].

We study the Cauchy problem for a class of strongly damped multidimensional generalized Boussinesq equations utt−Δu−Δutt+Δ2u+Δ2utt−kΔut=Δf(u), where k is a positive constant. Under some assumptions and by using potential well method, we prove the existence and nonexistence of global weak solution without solution without establishing the local exis...

This paper deals with the degenerate parabolic system ut=uΔu+u(a1−b1u+c1v)ut=uΔu+u(a1−b1u+c1v) and vt=vΔv+v(a2−b2v+c2u)vt=vΔv+v(a2−b2v+c2u) with homogeneous Dirichlet conditions in a bounded domain. We show that any positive solutions converge exponentially to the unique steady state if the coefficients satisfy certain conditions. The result is rel...

This paper is concerned with the Cauchy problem of solutions for some nonlinear multidimensional “good” Boussinesq equation of sixth order at three different initial energy levels. In the framework of potential well, the global existence and blowup of solutions are obtained together with the concavity method at both low and critical initial energy...

We study the initial boundary value problem of a class of fourth order semilinear parabolic equations. Global existence and nonexistence of solutions with initial data in the potential well are derived. Moreover, by using the iteration technique for regularity estimates, we obtain that for any k ≥ 0, the semilinear parabolic possesses a global attr...

This paper deals with large time behavior of the Dirichlet problem to the degenerate parabolic equation \({u_t = g(u) \Delta u + f(u)}\) in a bounded domain \({\Omega \subset R^n}\) with smooth boundary \({\partial \Omega}\) . Under suitable conditions on f(u) and g(u), we show that all solutions will converge to the steady state exponentially.

This paper considers the Cauchy problem of solutions for a class of sixth order 1-D nonlinear wave equations at high initial energy level. By introducing a new stable set we derive the result that certain solutions with arbitrarily positive initial energy exist globally.

This paper discusses the inhomogeneous nonlinear Schrödinger equation with critical exponent. By constructing a variational problem and so-called invariant manifolds of the evolution flow, we derive a sharp criterion for blow up and global existence of the solutions.

In this paper we study the initial-boundary value problem of the multidimensional viscoelasticity equation with nonlinear source term . By using the potential well method, we first prove the global existence. Then we prove that when time , the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial da...

With the method of system dynamics (SD), the model of the balance of water supply and demand in a region is set up. The amount of water demand can also be estimated by an exponential model. We also study the probability distributing of the precipitation. After all, we can predict water shortage in a certain region assisting people to design a bette...

In this paper, we consider three different convexity methods for proving blowup. First, we show principles of three different convexity methods by giving two lemmas. Then, by using respectively the three convexity methods we obtain the blowup result. Finally, we compare this three ways. This work aims to provide clear and comprehensive analysis for...

In this paper, we investigate the initial boundary value problem for nonlinear parabolic differential equations with both positive and negative absorption sources. We show that the solution of the above problem quenches in finite time and estimate its quenching time.

This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial en...

This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial en...

In this paper, we study the initial boundary value problem for fourth-order wave equations with nonlinear strain and source terms at high energy level. We prove that, for certain initial data in the unstable set, the solution with arbitrarily positive initial energy blows up in finite time.

We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.

We undertake a comprehensive study of the Cauchy problem of a class of second-order derivative nonlinear Schrödinger equations with combined power-type nonlinearities. Using the potential theory and the concavity method, we construct a variational problem and two invariant manifolds. Furthermore, we give sharp conditions of global existence and blo...

We study the initial boundary value problem for a class of fourth order strongly damped nonlinear wave equations u tt -Δu+Δ 2 u-αΔu t =f(u). By introducing a family of potential wells we prove the existence of global weak solutions and global strong solutions under some weak growth conditions on f(u). Furthermore we give the asymptotic behaviour of...

This article studies a nonlinear Schrödinger equation with harmonic potential by constructing different cross-constrained problems. By comparing the different cross-constrained problems, we derive different sharp criterion and different invariant manifolds that separate the global solutions and blowup solutions. Moreover, we conclude that some mani...

This paper concerns with the tripotency of a linear combination of three matrices, which has a background in statistical theory. We demonstrate all the possible cases that lead to the tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute. By utilizing block technique and returning the partitioned...

In this paper, we investigate the quenching phenomena of the initial boundary value problem for the fourth-order nonlinear parabolic equation in bounded domain. By some assumptions on the exponents and initial data for a class of equations with the general source term, we not only obtain the quenching phenomena in finite time but also estimate the...

In this paper, we investigate the quenching phenomena of the Cauchy problem for the second-order nonlinear parabolic equation on unbounded domain. It is shown that the solution quenches in finite time under some assumptions on the exponents and the initial data. Our main tools are comparison principle and maximum principle. We extend the result to...

We study the Cauchy problem of nonlinear Klein-Gordon equation with weak and strong damping terms. By employing the Galerkin method and Contraction Mapping Principle, we derive the local well-posedness of solutions.