Radial equivalence for the two basic nonlinear degenerate diffusion equations

In this paper we prove that there exists an explicit correspondence between the radially symmetric solutions of two well-known models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. We establish exact correspondence formulas between these solutions. We also study in detail the application of the results in the important case of self-similar solutions. In particular, we derive the existence of new self-similar solutions for the evolution p-Laplacian equation.     

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, t_∂u=Δpu, with 1<p<2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of Rn×[0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1<p?2n/(n+1). The boundedness results may be also extended to the limit case p=1, while the positivity estimates cannot.We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1<p<2, and point out their main properties.We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely .     

Multiple positive solutions of the one-dimensional p-Laplacian

In this work we investigate the existence of positive solutions of the p-Laplacian, using the quadrature method. We prove the existence of multiple solutions of the one-dimensional p-Laplacian for α?0, and determine their exact number for α=0.     

A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains

A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems −Δu=f(x,u) on a bounded domain Ω⊂Rn with u=0 on ∂Ω are studied, where the nonlinearity 0?f(x,s) grows at most like sp. If Ω is a Lipschitz domain we exhibit two exponents p_* and p^*, which depend on the boundary behavior of the Green function and on the smallest interior opening angle of ∂Ω. We prove that for 1<p<p_* all positive very weak solutions are a priori bounded in L^∞. For p>p^* we construct a nonlinearity f(x,s)=a(x)sp together with a positive very weak solution which does not belong to L^∞. Finally we exhibit a class of domains for which p_*=p^*. For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49-81].     

Very weak solutions with boundary singularities for semilinear elliptic Dirichlet problems in domains with conical corners

Let Ω⊂Rn be a bounded Lipschitz domain with a cone-like corner at 0∈∂Ω. We prove existence of at least two positive unbounded very weak solutions of the problem −Δu=up in Ω, u=0 on ∂Ω, which have a singularity at 0, for any p slightly bigger that the generalized Brezis-Turner exponent p^*. On an example of a planar polygonal domain the actual size of the p-interval on which the existence result holds is computed. The solutions are found variationally as perturbations of explicitly constructed singular solutions in cones. This approach also makes it possible to find numerical approximations of the two very weak solutions on Ω following a gradient flow of a suitable functional and using the mountain pass algorithm. Two-dimensional examples are presented.     

Positive solutions for one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative

This paper presents sufficient conditions for the existence and multiplicity of positive solutions to the one-dimensional p-Laplacian differential equation (?p^′(u^′))+a(t)f(u,u^′)=0, subject to some boundary conditions. We show that it has at least one or two or three positive solutions under some assumptions by applying the fixed point theorem.     

On self-similar solutions of semilinear wave equations in higher space dimensions

In this paper we analyze self-similar solutions of the semilinear wave equation Φ_tt − ΔΦ − Φ^p = 0 for n > 3 space dimensions. We found several classes of analytic solutions labeled by a single parameter, the form of which differ in the vicinity of the light cone. We also propose suitable numerical methods to study them.     

Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian

In this paper we consider the multiplicity of positive solutions for the one-dimensional p-Laplacian differential equation (?p^′(u^′))+q(t)f(t,u,u^′)=0, t∈(0,1), subject to some boundary conditions. By means of a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to some multipoint boundary value problems.     

A partial regularity result for an anisotropic smoothing functional for image restoration in BV space

Here we examine the partial regularity of minimizers of a functional used for image restoration in BV space. This functional is a combination of a regularized p-Laplacian for the part of the image with small gradient and a total variation functional for the part with large gradient. This model was originally introduced in Chambolle and Lions using the Laplacian. Due to the singular nature of the p-Laplacian we study a regularized p-Laplacian. We show that where the gradient is small, the regularized p-Laplacian smooths the image u, in the sense that u∈C1,α for some 0<α<1. This functional thus anisotropically smooths the image where the gradient is small and preserves edges via total variation where the gradient is large.     

Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ_* > 0 such that for λ < λ_*, the problem has no positive solution; for λ = λ_*, it has at least one positive solution; and for λ > λ_*, it has at least two positive solutions.     

Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.     

Dead-core rates for the fast diffusion equation with a spatially dependent strong absorption

This paper deals with the dead-core rates problem for the fast diffusion equation with a spatially dependent strong absorption $$u_t=(u^{m})_{xx}-x^{q}u^p, \quad(x,t)\in(0,1)\times(0,\infty),$$ where 0 < p < m < 1 and ?1 < q < 0. By using the self-similar transformation technique and the Zelenyak method, we proved that the temporal dead-core rate is non-self-similar.     

A multiplicity theorem for problems with the p-Laplacian

We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter λ∈R and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ₂=the second eigenvalue of , then the problem has at least three nontrivial solutions. Our approach combines the method of upper-lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p=2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990].     

Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α?max{m,p}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N=1, the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0?α<1, and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown.     

Regularity of the Interfaces with Sign Changes of Solutions of the One-Dimensional Porous Medium Equation

In previous papers we considered the Cauchy problem for the one-dimensional evolution p-Laplacian equation for nonzero, bounded, and nonnegative initial data having compact support, and showed that after a finite time the set of spatial critical points of the nonnegative solution u=u(x, t) in {u>0} consists of one point, the spatial maximum point of u, and the curve of the spatial maximum points is continuous with respect to the time variable. Since the spatial derivative ∂_xu satisfies the porous medium equation with sign changes, the curve of the spatial maximum points is regarded as an interface with sign changes of ∂_xu. On the other hand, in a paper by M. Bertsch and D. Hilhorst (1991, Appl. Anal.41, 111-130) the interfaces where the solutions change their sign were studied in detail for the initial-boundary value problems of the generalized porous medium equation over two-dimensional cylinders. But the monotonicity of the initial data is assumed there. As is noted in Section 4 of our earlier work (1996, J. Math. Anal. Appl.203, 78-103), the monotonicity of ∂_xu(?, t) in some neighborhood of the spatial maximum point of u(?, t) cannot be assumed, and therefore, if this monotonicity for some large t>0 is proved, then by the method of Bertsch and Hilhorst (cited above) one may get more precise regularity properties of the curve of the spatial maximum points. The purpose of the present paper is twofold. One is to remove some monotonicity assumption for initial data in Bertsch and Hilhorst's theorem concerning the regularity of the interfaces with sign changes of solutions of the one-dimensional generalized porous medium equation. By comparing the solution with appropriate symmetric nonnegative solutions we shall get the monotonicity of the solution near the interface after a finite time. The other is as a by-product of the method to get C¹ regularity of the curves of the spatial maximum points of nonnegative solutions of the Cauchy problem for the evolution p-Laplacian equation for sufficiently large t.     

Pointwise estimates for the fundamental solutions of a class of singular parabolic problems

We deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L ^∞ coefficients whose prototypes are the p-Laplacian (2N/(N + 1) < p < 2) and the porous medium equation (((N ? 2)/N)₊ < m < 1). We prove existence of and sharp pointwise estimates from above and from below for the fundamental solutions. Our results can be extended to general non-negative L ¹ initial data.     

Exact solutions of nonlinear dispersive K(m, n) model with variable coefficients

The variable-coefficient Korteweg-de Vries (KdV) equation with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condense was presented to describe the dynamics of nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. To understand the role of nonlinear dispersion in this variable-coefficient model, we introduce and study a new variable-coefficient KdV with nonlinear dispersion (called vc-K(m, n) equation). With the aid of symbolic computation, we obtain its compacton-like solutions and solitary pattern-like solutions. Moreover, we also present some conservation laws for both vc-K⁺(n, n) equation and vc-K⁻(n, n) equation.     

Some analytic approximations for neutral stochastic functional differential equations

We consider an analytic iterative method to approximate the solution of a neutral stochastic functional differential equation. More precisely, we define a sequence of approximate equations and we give sufficient conditions under which the approximate solutions converge with probability one and in pth moment sense, p ? 2, to the solution of the initial equation. We introduce the notion of the Z-algorithm for this iterative method and present some examples to illustrate the theory. Especially, we point out that the well-known Picard method of iterations is a special Z-algorithm.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.