Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

Asymptotics of solutions of nonlinear parabolic equations

This paper is concerned with the large time behaviour of solutions to the Cauchy problem of the following nonlinear parabolic equations:

     

Classical solutions of fully nonlinear parabolic equations

In this paper, we study the classical solutions of the fully nonlinear parabolic equation u_t-F(D_x²u)=0,{u_{t}-F(D_{x}^2u)=0,} where the nonlinear operator F is locally C ^1,β almost everywhere with 0 < β < 1. The interior C ^2,α regularity of the classical solutions will be shown without the assumption that F is convex (or concave).     

Extinction of solutions of parabolic equations with variable anisotropic nonlinearities

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions that generalize the evolutional p(x, t)-Laplacian. We study the property of extinction of solutions in finite time. In particular, we show that the extinction may take place even in the borderline case when the equation becomes linear as t → ∞.     

Verification of asymptotic solutions for one-dimensional nonlinear parabolic equations

The present article proves a result that is new for partial differential equations. According to this result, the solution of the Cauchy problem for a nonlinear parabolic equation with variable, slowly changing coefficients will turn into (asymptotically approach) a special asymptotic solution, either a solution or a kink, for high values of t.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 10–16, September, 1992.     

On viscosity solutions for nontotally parabolic fully nonlinear equations

It is shown how to prove global unique solvability of the first initial-boundary value problem in the class of continuous viscosity solutions for some classes of equations −u_t+F(u_x,u_xx)=g(x, t, u_x), where F(p, A) is elliptic only on some nonlinear subsets of values of the arguments (p, A). For this purpose we use the techniques developed in the theory of viscosity solutions for degenerate elliptic equations. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 112–130.     

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.