Chunpeng Wang

Jilin University | JUT ·  Department of Mathematics and Applied Mathematics

This paper concerns the asymptotic behavior of solutions to one-dimensional coupled semilinear degenerate parabolic equations with superlinear reaction terms both in bounded and unbounded intervals. The equations are degenerate at a lateral boundary point and the diffusion coefficients are general functions. For the problem in a bounded interval, i...

This paper concerns subsonic-sonic potential flows in general two dimensional nozzles. For finitely long symmetric nozzles, we formulate the subsonic-sonic flow problem by prescribing the flow angle at the inlet and the outlet. It is shown that this problem admits a unique Lipschitz continuous subsonic-sonic flow, and the sonic points of the flow m...

This paper concerns continuous subsonic-sonic potential flows in a two dimensional convergent nozzle, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. It is shown that for a given nozzle which is a perturbation of an straight one, and a given mass flux, there exists uniquely a continuous subsonic-sonic flo...

A smooth transonic flow problem is formulated as follows: for a de Laval nozzle, one looks for a smooth transonic flow of Meyer type whose sonic points are all exceptional and whose flow angle at the inlet is prescribed. If such a flow exists, its sonic curve must be located at the throat of the nozzle and the nozzle should be suitably flat at its...

This paper deals with two-dimensional subsonic-sonic potential flows in general nozzles, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. For a large class of nozzles, it is shown that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique subsonic-sonic flows in...

This paper concerns compressible subsonic jet flows for a given surrounding pressure from a two-dimensional finitely long convergent nozzle with straight solid wall, which are governed by a free boundary problem for a quasilinear elliptic equation. For a given surrounding pressure and a given incoming mass flux, we seek a subsonic jet flow with the...

This paper concerns a control system governed by a convection-diffusion equation, which is weakly degenerate at the boundary. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. The Carleman estimate is established by means of a suitable transformation, by which the d...

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equatio...

This paper concerns the Neumann problem of a reaction-diffusion system, which has a variable exponent Laplacian term and could be applied to image denoising. It is shown that the problem admits a unique renormalized solution for each integrable initial datum.

This paper concerns the Neumann problem of a reaction-diffusion system, which has a variable exponent Laplacian term and could be applied to image denoising. It is shown that the problem admits a unique renormalized solution for each integrable initial datum.

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle with straight solid walls. It is shown that for the given inlet, which is a perturbation of an arc centered at the vertex of the nozzle, and the given incoming flow angle which is a perturbation of the angle of the inner normal of the inlet, and the...

This paper concerns properties of sonic curves for two-dimensional smooth subsonicsonic and transonic steady potential flows, which are governed by quasi-linear degenerate elliptic equations and elliptic-hyperbolic mixed-type equations with degenerate free boundaries, respectively. It is shown that a sonic point satisfying the interior subsonic cir...

This paper concerns continuous subsonic-sonic potential flows in a two dimensional finite nozzle with a general upper wall and a straight lower wall. We give a class of nozzles where continuous subsonic-sonic flows may exist. Consider a continuous subsonic-sonic flow in such a nozzle after rescaling the upper wall in a small scale. It is shown that...

This paper concerns the quenching phenomenon of solutions to a class of semilinear parabolic equations with boundary degeneracy. In the case that the degeneracy is not strong, it is shown that there exists a critical length, which is positive, such that the solution exists globally in time if the length of the spatial interval is less than it, whil...

This paper concerns smooth supersonic flows with Lipschitz continuous speed in two-dimensional infinite expanding nozzles, which are governed by a quasilinear hyperbolic equation being singular at the sonic and vacuum state. The flow satisfies the slip condition on the walls and the flow velocity is prescribed at the inlet. First, it is proved that...

This paper concerns a control system governed by a semilinear degenerate equation involving a fully nonlinear gradient term. The equation may be weakly degenerate and strongly degenerate on a portion of the lateral boundary, and the gradient term can be controlled by the diffusion term. The linearized system is shown to be approximately controllabl...

This paper concerns a class of control systems governed by semilinear degenerate parabolic equations with convection terms, which cannot be controlled by the diffusion terms. The Carleman estimates and the observability inequalities for the corresponding linear equations are established if the degeneracy is relatively weak. Subsequently, it is prov...

This paper is devoted to the first initial boundary value problems of a class of forward-backward convection-diffusion equations. The existence theorem and the continuous dependence theorem of Young measure solutions are established.

This paper concerns the asymptotic behavior of solutions to a semilinear parabolic equation with boundary degeneracy. It is proved that for the problem in a bounded domain with a homogeneous boundary condition, there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong, while the nontrivial solution must blow up in...

This paper concerns the well-posedness of a boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary. Such problems arise when studying continuous subsonic–sonic flows in a convergent nozzle with straight solid walls. It is shown that for a given inlet being a perturbation of an arc centered at...

This article concerns the quenching phenomenon of the solution to the Dirichlet problem of a singular and degenerate quasilinear diffusion equation. It is shown that there exists a critical length for the special domain in the sense that the solution exists globally in time if the length of the special domain is less than this number while the solu...

This paper concerns a class of control systems governed by semilinear degenerate equations with convection term. The equations may be weakly degenerate and strongly degenerate on a portion of the lateral boundary, and the convection terms can be controlled by the diffusion terms. The control systems are shown to be approximately controllable.

This paper studies a class of weighted non-Newtonian filtration equations with slow diffusion. By using the method introduced by Galaktionov and Levine for the classical non-Newtonian filtration equation, we establish the blow-up theorems of Fujiita type for the extended model, where more difficult and complicated estimates are required to treat th...

This paper concerns smooth transonic flows of Meyer type in finite de Laval nozzles, which are governed by an equation of mixed type with degeneracy and singularity at the sonic state. First we study the properties of sonic curves. For any $C^2$ transonic flow of Meyer type, the set of exceptional points is shown to be a closed line segment (may be...

This paper concerns the null controllability of the system governed by coupled degenerate equations. By the Carleman estimate for the case of a single degenerate equation, the Carleman estimate and the observability inequality are established. Then, the system with two controls and the system with one control are shown to be null controllable.

In the past, we established a module structure theorem for Sobolev spaces on open manifolds with bounded curvature and positive injectivity radius r inj (M)=inf x∈M r inj (x)>0. The assumption r inj (M)>0 was essential in the proof. But, manifolds (M n ,g) with vol(M n ,g)<∞ have been excluded. An extension of our former results to the case vol(M n...

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Hölder estimates for weak solutions by the estimates in the Campanato space. Based on such Hölder esti-mates, we then investigate subsonic-sonic flows with s...

This paper concerns a convective nonlinear diffusion equation which is strongly degenerate. The existence and uniqueness of the BV solution to the initial-boundary problem are proved. Then we deal with the anti-shifting phenomenon by investigating the corresponding free boundary problem. As a consequence, it is possible to find a suitable convectio...

This paper deals with the large-time behaviour of solutions to the exterior problem of the non-Newtonian filtration equation with first-order term and nonlinear boundary source. In particular, the critical global exponent and the critical Fujita exponent are determined or estimated. An interesting phenomenon is shown: there exists a threshold value...

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 stil...

In this paper we consider a singular diffusion equation arising in phase transition and investigate its self-similar entropy solutions with jump hypersurfaces. The existence, nonexistence and uniqueness theorems of such solutions are established. We also discuss some properties of this kind of solutions including monotonicity and asymptotic behavio...

In this paper we consider the approximate controllability of a class of degenerate systems. The equations may be weakly degenerate and strongly degenerate on the boundary. We prove that the control systems are approximate controllable by constructing the controls via the conjugate problems.

In this paper, we investigate the large time behaviour of solutions to the exterior problems of a class of quasilinear parabolic equations with convection terms. We establish the critical Fujita exponents pc and blow-up theorems of the Fujita type for both homogeneous Neumann and Dirichlet problems. In particular, it is shown that the critical p =...

The paper concerns the existence theorem of weak solutions for the initial-boundary value problem of a nonlinear diffusion equation with convection. The equation may be regarded as a generalized non-Newtonian polytropic filtration equation. By doing the necessary BVBV estimate and other estimates for approximating solutions, we establish the existe...

This paper is concerned with the heat equation with a reaction source of spatio-temporal delay. Special attention is paid to the determination of critical exponents used to describe large time behaviour of solutions of the Cauchy problem.

In this paper we consider the approximate controllability of a class of degenerate semilinear systems. The equations may be weakly degenerate and strongly degenerate on a portion of the lateral boundary. We prove that the control systems are approximately controllable and the controls can be taken to be of quasi bang-bang form. Mathematics Subject...

This paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem. The equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies. By using Holmgren’s approach, we prove that the bounded solution of the associated initial–boundary value p...

This paper deals with the exterior problem of the Newtonian filtration equation with nonlinear boundary sources. The large time behavior of solutions including the critical Fujita exponent are determined or estimated. An interesting phenomenon is illustrated that there exists a threshold value for the coefficient of the lower order term, which depe...

We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infin...

This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison pri...

We study the self-similar solutions and the time-asymptotic behaviour of solutions for a class of degenerate and singular diffusion equations in the form u t =(|(p(u)) x | λ-2 (p(u)) x ) x ,-∞<x<+∞,t>0, where λ>2 is a constant. The existence, uniqueness and regularity for the self-similar solutions are obtained. In particular, the behaviour at two...

The paper concerns the well-posedness problem of an evolutionary weighted p-Laplacian with boundary degeneracy. Different from the classical theory for linear equations, it is shown that the degenerate portion of the boundary should be decomposed into two parts: the strongly degenerate boundary on which the equation exhibits hyperbolic characterist...

In this paper, we establish the blow-up theorems of the Fujita type for a class of homogeneous Neumann problems of quasilinear equations with convection terms. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow-up case under any nontrivial initial data. An interesting phenomenon is exploited such that...

This work is concerned with the critical exponent of the non-Newtonian polytropic filtration equation with nonlinear boundary conditions. We obtain the critical global existence exponent and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions.

This book provides an introduction to elliptic and parabolic equations. While there are numerous monographs focusing separately on each kind of equations, there are very few books treating these two kinds of equations in combination. This book presents the related basic theories and methods to enable readers to appreciate the commonalities between...

In this paper, we consider the initial-boundary value problem of a nonlinear parabolic equation with double degeneracy, and establish the existence and uniqueness theorems of renormalized solutions which are stronger than BV solutions.

In this paper we investigate the critical Fujita exponent for the initial-value problem of the degenerate and singular nonlinear parabolic equation vertical bar x vertical bar(lambda 1) (partial derivative u)/(partial derivative t) = Delta u(m) + vertical bar x vertical bar(lambda 2)u(p), + x is an element of R-n, t > 0, with a non-negative initial...

In this paper, we study the similar entropy solutions of the singular diffusion equation, ∂u∂t=∂∂xψ(∂u∂x), with ψ(s)=s/1+s2. These kinds of solutions have nonvertical jump lines. We establish the existence and uniqueness and also discuss some properties of these kinds of solutions.

Radially symmetric solutions of the p-Laplacian were considered in the perforated-like domain with nonlocal boundary conditions. Singular ordinary differential equations were important for the applications in physical sciences such as the study of non-Newtonian fluid theory and the turbulent flow of a gas in a porous medium. The results were not a...

In this paper we study self-similar solutions to the singular and degenerate diffusion equationwhere 1<λ<2. The existence and uniqueness for the solutions are established. In addition, the asymptotic behavior is investigated.

We consider the viscous Cahn-Hilliard equation with spatial dimension n≤5, and establish global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial data.

In this paper we study the convection diffusion equation ∂u/∂t = Δm - x ·∇uq, (x,t) ∈ ℝn × (O, + ∞), where m > 1, 1 < q ≤ m. We are interested in similar solutions with the properties of finite speed propagation of perturbations and with shrinking or unchanged supports. We establish the existence and uniqueness, and then discuss some properties of...

In this paper we study the shrinking self-similar solutions of the nonlinear diffusion equation with nondivergence form ∂u∂t=umΔu(m⩾1). This kind of solutions possess the properties of finite speed propagation of perturbations and their supports are shrinking. We establish the existence and uniqueness for this kind of solutions. In addition, we stu...

This paper is devoted to Young measure solutions of a class of forward–backward diffusion equations. Inspired by the idea from a recent work of Demoulini, we first discuss the regular case by introducing the Young measure solutions and prove the existence for such solutions, and then approximate the extreme case by the approach of regularization an...

This paper is concerned with the first initial-boundary value problem of a class of singular diffusion equations with the flux sublinear growth and the potential without convexity, in which the properties of Young measure solutions are discussed, and the uniqueness, stability are proved, and the asymptotic behavior of solutions and their energy as...

In this paper we study the approximate controllability of a class of quasilinear parabolic equations in a bounded spacial domain Ω⊂ℝ N when the control acts on any open and nonempty subset of Ω. The approximate controllability in L p (Ω) for N+2≤p<+∞ is proved. The proof combines a variational approach to the controllability problem for linear equa...

The first initial-boundary value problem of a class of singular diffusion equations with the flux sublinear growth and the potential without convexity is investigated. Such equations may be strongly degenerate, singular and forward-backward. Inspired by the idea in a recent work of Demoulini, we first discuss the regular case by introducing the You...

In this paper we study the uniqueness of solutions to the initial and Dirichlet boundary-value problem of dierential inclusions

The authors investigate the following nonlinear diffusion equations with boundary degeneracy and gradient nonlinearity ∂u ∂t-diva(x)|∇u| p-2 ∇ u+b i (x)D i u+c(x,t)u=f(x,t),(x,t)∈Ω×(0,T), where repeated indices denotes the summation from 1 to N, Ω⊂ℝ N is a bounded domain with appropriately smooth boundary ∂Ω, p>1, a∈C(Ω ¯) and a(x)>0 for x∈Ω. Such...

We construct a single transonic shock wave pattern in an infinite curved nozzle with decay cross-section, which is close to a unform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in a non-curved nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a litt...