Effects of reactive gradient term in a multi-nonlinear parabolic problem

This paper deals with parabolic equation ut=Δu+r|∇u|−aepu subject to nonlinear boundary flux ∂u/∂η=equ, where r>1, p,q,a>0. There are two positive sources (the gradient reaction and the boundary flux) and a negative one (the absorption) in the model. It is well known that blow-up or not of solutions depends on which one dominating the model, the positive or negative sources, and furthermore on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of the reactive gradient term on the asymptotic behavior of solutions. We at first determine the critical blow-up exponent, and then obtain the blow-up rate, the blow-up set as well as the spatial blow-up profile for blow-up solutions in the one-dimensional case. It turns out that the gradient term makes a substantial contribution to the formation of blow-up if and only if r?2, where the critical r=2 is such a balance situation of the two positive sources for which the effects of the gradient reaction and the boundary source are at the same level. In addition, it is observed that the gradient term with r>2 significantly affects the blow-up rate also. In fact, the gained blow-up rates themselves contain the exponent r of the gradient term. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. As for the influence of gradient perturbations on spatial blow-up profiles, we only need some coefficients related to r for the profile estimates, while the exponent of the profile itself is r-independent. This seems natural for boundary blow-up solutions that the spatial profiles mainly rely on the exponent of the boundary singularity.     

Rate estimates of gradient blowup for a heat equation with exponential nonlinearity

We consider a one-dimensional semilinear parabolic equation , for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.     

Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity

We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with supercritical nonlinearity. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its sharp convergence rate for a class of initial data. We also derive a universal lower bound of the convergence rate which implies the optimality of the result. Proofs are given by a comparison method based on matched asymptotics expansion.     

Support properties of solutions to a degenerate equation with absorption and variable density

In this paper, we study the properties of solutions to a degenerate parabolic equation with variable density and absorption. We first obtain a critical exponent, which distinguishes the localization of solutions from the positivity of them. When positivity prevails, we obtain the other critical exponent with respect to the decay of the variable density, which separates the global existence of interfaces from the disappearance of them. Moreover, the long time behavior of interfaces is characterized.     

A note on semilinear heat equation in hyperbolic space

Bandle et al. [1] obtained a quite interesting result about a semilinear heat equation that the Fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in Euclidean space, and, in addition, the properties of solutions are different in the critical exponent case. Our purpose is to answer an open problem proposed by Bandle et al. for the critical exponent case, and it, together with the one obtained by them, shows that the critical exponent case does belong to the non-blow-up case, which is completely different from the case in Euclidean space.     

Exterior nonlinear Neumann problem

We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical. We establish the existence of least energy solutions. In the subcritical case the coefficient b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.     

Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay

We consider a reaction-diffusion system with general time-delayed growth rate and kernel functions. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained. Moreover, taking minimal time delay τ as the bifurcation parameter, Hopf bifurcation near the steady-state solution is proved to occur at a critical value τ=τ₀. Especially, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to competitive and cooperative systems with weak or strong kernel function respectively.     

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.     

Linking method applied to a class of a system of critical growth wave equations

We show the existence of at least two solutions for a class of a system of critical growth wave equations, with periodic condition on t and the Dirichlet boundary condition

     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

On the regularity criteria of a surface quasi-geostrophic equation

We obtain regularity criteria for a quasi-geostrophic equation that depends more on one direction than the others. In particular, we show that in the critical case, the global regularity depends only on a partial derivative rather than a gradient of the solution.     

Small amplitude steady internal waves in stratified fluids

We study the weakly non linear solutions of theDubreil-Jacotin—Long elliptic equation in a strip, which describes two dimensional gravity internal waves propagating steadily in a stratified fluid. In the neighborhood of the first critical value of the Froude number, the center manifold theorem ensures that small solutions are parametrized by two coordinates which verify a system of nonlinear ordinary differential equations. We compute numerically the coefficients of the normal form of this reduced system for a three parameters family of stratifications and show that the quadratic coefficient (the most important) may become small. In that case, nonusual waves such as fronts can propagate. The last part of our work studies the case when a smooth stratification converges towards a piecewise constant profile having one discontinuity. We observe formally that the small waves which propagate at the interface of two homogeneous fluids are limits at leading order of waves travelling in the region where the smooth density varies rapidly.     

Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annulus

Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.     

Boundedness of solutions to the critical fully parabolic quasilinear one‐dimensional Keller–Segel system

In this paper we consider a one‐dimensional fully parabolic quasilinear Keller–Segel system with critical nonlinear diffusion. We show uniform‐in‐time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well‐known Lyapunov functional and a recently introduced new Lyapunov‐like functional in 3 .     

Cauchy problems of semilinear pseudo-parabolic equations

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.     

Phase plane analysis and classification of solutions of a mixed sublinear-superlinear elliptic problem

In this paper, some mixed sublinear-superlinear critical problem extending the famous problem of Brezis–Nirenberg are analysed. The existence of solutions is discussed. A phase plane analysis is performed in order to transform the problem into an ordinary differential equation. Finally, a full classification of radial solutions according to their behavior at the origin is performed for subcritical, critical and supercritical cases.     

Asymptotics for a free boundary model in price formation

We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria.The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.