Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as _∂ttu−Δu=f(x,u) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as t_∂u−Δu=f(x,u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.     

Uniqueness properties of higher order dispersive equations

In this paper we study unique continuation properties of solutions to higher (fifth) order nonlinear dispersive models. The aim is to show that if the difference of two solutions of the equations, u₁−u₂, decays sufficiently fast at infinity at two different times, then u₁≡u₂.     

Double reduction of a nonlinear (2+1) wave equation via conservation laws

Conservation laws of a nonlinear (2+1) wave equation u_tt = (f(u)u_x)_x +  (g(u)u_y)_y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f′(u) and g′(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f′(u) and g′(u) are linearly independent.     

Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary

This paper establishes the blowup rate estimate near the blowup time for the heat equation u_t=Δu with the nonlinear boundary condition u_n=u^p on the boundary ∂Ω×[0,T) for a bounded Lipschitz domain and for the nonlinear boundary condition u_n=exp(u) on the boundary for a two-dimensional disc with nonradially symmetric data.     

Existence results for a class of second order evolution inclusions and its corresponding first order evolution inclusions

We deal with an abstract second order nonlinear evolution inclusion with its principal part having a small parameter ?. We prove the existence of a weak solution when the nonlinearity F is convex as well as nonconvex valued. Then we study the asymptotic behavior of a sequence of solutions {u _? } when ? → 0. We prove that there exists a limit function u, and u is a solution of the corresponding first order evolution inclusion.     

The focusing energy-critical Hartree equation

We consider the focusing energy-critical nonlinear Hartree equation iut+Δu=−(⁻⁴|x|∗²|u|)u. We proved that if a maximal-lifespan solution u:I×Rd→C satisfies supt∈I‖∇u(t)_‖2<‖∇W_‖2, where W is the static solution of the equation, then the maximal-lifespan I=R, moreover, the solution scatters in both time directions. For spherically symmetric initial data, similar result has been obtained in [C. Miao, G. Xu, L. Zhao, Global wellposedness, scattering and blowup for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., in press]. The argument is an adaptation of the recent work of R. Killip and M. Visan [R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrödinger equations.     

Higher order of fully discreate solution for parabolic problem inL ∞

In this work, we approximate the solution of initial boundary value problem using a Galerkin-finite element method for the spatial discretization, and Implicit Runge-Kutta methods for the time stepping. To deal with the nonlinear termf(x, t, u), we introduce the well-known extrapolation sheme which was used widely to prove the convergence inL ₂-norm. We present computational results showing that the optimal order of convergence arising underL ₂-norm will be preserved inL _∞-norm.     

Global existence of solutions for semilinear damped wave equation in 2-D exterior domain

We consider a mixed problem of a damped wave equation u_tt−Δu+u_t=|u|^p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term |u|^p satisfies p∗=2<p<+∞. For this purpose we shall deal with a radially symmetric solution in the exterior domain. A new device developed in Ikehata-Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.     

Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems

We prove existence, uniqueness, regularity results and estimates describing the behavior (both for large and small times) of a solution u of some nonlinear parabolic equations of Leray-Lions type including the p-Laplacian. In particular we show how the summability of the initial datum u₀ and the value of p influence the behavior of the solution u, producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates.     

A practical choice of parameters in improved SOR-Newton method with orderings

In this paper, on the basis of the results of Ishihara et al. (1997), we first discuss global convergence theorems for the improved SOR-Newton and block SOR-Newton methods with orderings applied to a system of mildly nonlinear equations, which includes as a special case the discretized version of the Dirichlet problem, for the equation ϵΔu + p(x)u_x + q(y)u_y = f(x, y, u), where f is continuously differentiable and fu(x, y, u) ⩾ 0. Moreover, we propose a practical choice of the multiple relaxation parameters {ω_i} for them. Numerical examples are also given.     

Obstructions to Existence in Fast-Diffusion Equations

The study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory. Thus, in the sub-field of very fast diffusion it is known that the Cauchy problem can be ill-posed, either because of non-uniqueness, or because of non-existence of solutions with small data. The equations we consider take the general form u_t=(D(u,u_x)u_x)_x or its several-dimension analogue. Fast diffusion means that D→∞ at some values of the arguments, typically as u→0 or u_x→0. Here, we describe two different types of non-existence phenomena. Some fast-diffusion equations with very singular D do not allow for solutions with sign changes, while other equations admit only monotone solutions, no oscillations being allowed. The examples we give for both types of anomaly are closely related. The most typical examples are v_t=(v_x/∣v∣)_x and u_t=u_xx/∣u_x∣. For these equations, we investigate what happens to the Cauchy problem when we take incompatible initial data and perform a standard regularization. It is shown that the limit gives rise to an initial layer where the data become admissible (positive or monotone, respectively), followed by a standard evolution for all t>0, once the obstruction has been removed.     

On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

In this paper we study the initial problem for a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws. The equation is driven by Lévy space-time white noise in the following form:

(t_∂−A)u+x_∂q(u)=f(u)+g(u)Ft,x     

On an over-determined problem of free boundary of a degenerate parabolic equation

This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants K and T ₀, to decide the initial value u ₀ such that the solution u(x, t) satisfies $\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K$ , where H _u(T ₀) = {x, ?^N: u(x, T ₀) > 0}. In this paper, we first establish a priori estimate u _t ? C(t)u and a more precise Poincaré type inequality $\left\| \phi \right\|_{L^2 (B_\varrho )}^2 \leqslant \varrho \left\| {\nabla \phi } \right\|_{L^2 (B_\varrho )}^2 $ , and then, we give a positive constant C ₀ and assert the main results are true if only $\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0 $ .     

Inverse problem of groundwater modeling by iteratively regularized Gauss-Newton method with a nonlinear regularization term

A nonlinear minimization problem ‖F(d)−u‖?min, ‖u−u_δ‖≤δ, is a typical mathematical model of various applied inverse problems. In order to solve this problem numerically in the lack of regularity, we introduce iteratively regularized Gauss-Newton procedure with a nonlinear regularization term (IRGN-NRT). The new algorithm combines two very powerful features: iterative regularization and the most general stabilizing term that can be updated at every step of the iterative process. The convergence analysis is carried out in the presence of noise in the data and in the modified source condition. Numerical simulations for a parameter identification ill-posed problem arising in groundwater modeling demonstrate the efficiency of the proposed method.     

Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball

In this paper we study the uniqueness and nondegeneracy of positive solutions of nonlinear problems of the type Δ_pu+f(r,u)=0 in the unit ball B, u=0 on ∂B. Here Δ_p denotes the p Laplace operator Δ_p=div(|∇u|^p−2∇u), p>1. The main ideas rely on the Maximum Principle and an implicit function theorem that we derive in a suitable weighted space. This space is essential to deal with the case p≠2.     

Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation

The nonlinear wave equation utt=(c²x(u)ux) arises in various physical applications. Ames et al. [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] did the complete group classification for its admitted point symmetries with respect to the wave speed function c(u) and as a consequence constructed explicit invariant solutions for some specific cases. By considering conservation laws for arbitrary c(u), we find a tree of nonlocally related systems and subsystems which include related linear systems through hodograph transformations. We use existing work on such related linear systems to extend the known symmetry classification in [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] to include nonlocal symmetries. Moreover, we find sets of c(u) for which such nonlinear wave equations admit further nonlocal symmetries and hence significantly further extend the group classification of the nonlinear wave equation.     

Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order

This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: ut+∇⋅(|∇Δu|^p−2∇Δu)=f(u) in Ω⊂RN with boundary condition u=Δu=0 and initial data u₀. The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u₀ as p→∞.     

On bounds for solutions to nonlinear wave equations in Hilbert space with applications to nonlinear elastodynamics

For the nonlinear wave equationu _tt -Nu +G(t,u, u _t ) = ? in Hilbert space, with associated homogeneous initial data, we show how ana priori bound of the form ∫ ₀ ^T ∥G(τ,u, u _τ)∥² dτ ≤ κ ∫ ₀ ^T ∥?(τ)∥² dτ leads to upper and lower bounds for ∥u∥ in terms of ∥?∥. An application to nonlinear elastodynamics is presented.     

Nonlinear abstract differential equations with deviated argument

In this paper we consider the nonlinear differential equation with deviated argument u^′(t)=Au(t)+f(t,u(t),u[φ(u(t),t)]), t∈R₊, in a Banach space (X,‖⋅‖), where A is the infinitesimal generator of an analytic semigroup. Under suitable conditions on the functions f and φ, we prove a global existence and uniqueness result for the above equation.     

Generalized solutions to nonlinear first-order systems

In the present work we reinterpret a result of approximate solutions to a nonlinear first order system in the framework of Colombeau's theory, defining an algebra of generalized germs where the problem $$u_t = f(x,t,u,u_x ), u|_{t = 0} = u_o (x)$$ has a unique solution, wheref andu ₀ are vector-valued functions.