Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.     

Pointwise Green's function estimates toward stability for multidimensional fourth-order viscous shock fronts

For the case of multidimensional viscous conservation laws with fourth-order smoothing only, we develop detailed pointwise estimates on the Green's function for the linear fourth-order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution. As in previous analyses in the case of second-order smoothing, our estimates are sufficient to establish that spectral stability implies nonlinear stability, though the full development of this result will be considered in a companion paper.     

Symmetry analysis,conserved quantities and applications to a dissipative DGH equation

In this paper, we study a weakly dissipative Dullin–Gottwald–Holm equation from the viewpoint of Lie symmetry analysis. We first perform symmetry analysis and the nonlinear self-adjointness of this equation. Due to a mixed derivatives term in the equation, we need to rewrite the corresponding form Lagrangian in symmetric form to construct conservation laws. From the viewpoint, we present a general procedure of how these conserved quantities come about. Based on these conserved quantities, blow-up analysis and global existence of strong solutions are presented. Finally, we show that this equation admits a weak peakon-type solution.     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Viscosity solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplace equation with nonlinear source

In the present paper we consider the Dirichlet problem in a convex domain for the multidimensional p-Laplace equation with nonlinear source. We prove the existence of the unique continuous viscosity solution under quite general assumptions on the structure of the source. Received: 18 January 2006     

On the uniqueness and structure of solutions to a coupled elliptic system

In this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball. In addition, the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved.     

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

Ricci solitons: the equation point of view

We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.     

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities.     

Conservation laws for a modified lubrication equation

In this work we study the conservation laws of a modified lubrication equation, which describes the dynamics of the interfacial motion in phase transition. We show that the equation is nonlinear self-adjoint and has an exact Lagrangian with an auxiliary function. As a result, by a general theorem on conservation laws proved by Nail Ibragimov recently and Noether’s theorem, some new conservation laws for the equation are obtained. Our results show that the non-locally defined conservation laws generated by Noether’s theorem are equivalent to the local ones given by Ibragimov’s theorem.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.     

Nonlinear self-adjointness and conservation laws for a generalized Fisher equation

In this work we study a generalization of the well known Fisher equation. We determine the subclasses of these equations which are nonlinear self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov and the symmetry generators we find conservation laws for these partial differential equations without classical Lagrangians.     

Nonlinear self-adjointness through differential substitutions

It is known (Ibragimov, 2011; Galiakberova and Ibragimov, 2013) [14,18] that the property of nonlinear self-adjointness allows to associate conservation laws of the equations under study, with their symmetries. In this paper we show that, even when the equation is nonlinearly self-adjoint with a non differential substitution, finding the explicit form of the differential substitution can provide new conservation laws associated to its symmetries. By using the general theorem on conservation laws (Ibragimov, 2007) [11] and the property of nonlinear self-adjointness we find some new conservation laws for the modified Harry-Dym equation. By using a differential substitution we construct a conservation law for the Harry-Dym equation, which has not been derived before using Ibragimov method.     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Subsonic solutions for compressible transonic potential flows

We establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely. It has been conjectured by experiments and numerical studies that the self-similar multidimensional flow changes its type, namely, hyperbolic far from the origin (supersonic region) and elliptic near the origin (subsonic region). Furthermore, the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation, the nonlinear wave equation, and the pressure gradient equation. Namely, the coefficients of the potential equation depend on the gradients while others are independent of the gradients. We provide techniques to handle the gradients, establish interior and boundary gradient estimates for the potential flow in a convex region, and answer the conjecture, that is, the flow is strictly elliptic and the region is subsonic.     

A uniform bound for the solutions to a simple nonlinear equation on Riemannian manifolds

Let M be a complete noncompact manifold with Ricci curvature bounded below. In this note, we derive a uniform bound for the solutions to the nonlinear equation

     

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.     

A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation

We present the existence of the subsonic solution to a two-dimensional Riemann problem governed by a self-similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle. Thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle. By utilizing the barrier methods and iterative methods, we show the well-posedness of the transonic shock in the entire subsonic region and thus establish the global solution. This result does not rely on any smallness of Riemann data.