The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C¹ solutions with the bounded L¹∩L^∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C¹ travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R^1+(1+n).     

Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. Based on the existence results of global classical solutions, we prove that when t tends to infinity, the solution approaches a combination of C¹ travelling wave solutions, provided that L¹ ∩ L^∞ norm of the initial data as well as its derivative are bounded. Application is given for the time‐like extremal surface in Minkowski space. Copyright © 2006 John Wiley & Sons, Ltd.     

Extremal maps in Sobolev type inequalities: Some remarks

In this work we make some observations on the existence of extremal maps for sharp L²-Riemannian Sobolev type inequalities as Nash and logarithmic Sobolev ones. Among other results, we prove also that there exist smooth compact Riemannian manifolds with scalar curvature changing signal on which there exist extremal maps.     

A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions

We study the comparison principle for degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and supersolution. The L¹ contractivity and, therefore, uniqueness of entropy solutions has been obtained so far by some authors, but it seems that any comparison theorem is not proven. The method used there is the doubling variable technique due to Kru?kov. Our method is based upon the kinetic formulation and the kinetic techniques. By developing the kinetic techniques for degenerate parabolic-hyperbolic equations with boundary conditions, we can obtain a comparison property which obviously extends the L¹ contractive property.     

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.     

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L ^γ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in L ^γ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence.     

Large time behavior and L−L estimate of solutions of 2-dimensional nonlinear damped wave equations

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L^p−L^q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L^p−L^q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|^αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2.     

Regularity of radial minimizers and extremal solutions of semilinear elliptic equations

We consider a special class of radial solutions of semilinear equations −Δu=g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, Lq, and Wk,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution is bounded if n?9 (for every g), and belongs to H³=W3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent.     

The Cauchy problem for critical and subcritical semilinear parabolic equations in Lr (II), initial data in critical Sobolev spaces Ḣ−s,rs

By presenting some time–space L^p−L^r estimates, we will establish the local and global existence and uniqueness of solutions for semilinear parabolic equations with the Cauchy data in critical Sobolev spaces of negative indices. Our results contain the complex (derivative) Ginzburg–Landau equation and the Cahn–Hilliard equation as special cases.     

Non-monotone stochastic generalized porous media equations

By using the Nash inequality and a monotonicity approximation argument, existence and uniqueness of strong solutions are proved for a class of non-monotone stochastic generalized porous media equations. Moreover, we prove for a large class of stochastic PDE that the solutions stay in the smaller L²-space provided the initial value does, so that some recent results in the literature are considerably strengthened.     

A local property ofL-regular sets

A compact subsetK of the unit ball in ? ⁿ is said to beL-regular if the extremal function $$h_K (z) = \sup \{ f(z):f \in PSH,f< 0f \leqslant - 1 on K\} $$ is continuous. With a method from an earlier paper we prove thatL-regularity is essentially a local property of the setK. As a corollary we note that, in the special case when ? ⁿ ,L-regularity is nothing but a local property ofK.     

Perturbed S1-symmetric hamiltonian systems

By techniques of critical point theory, we show that multiple periodic weak solutions of a general class of Hamiltonian systems persist despite perturbation with an L² term destroying the S¹-symmetries.     

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic- parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ Ll-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.     

On well-posedness of the semilinear heat equation on the sphere

We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L ^p ?? L ^q estimates for the semigroup generated by the Laplace?CBeltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n??? 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Stefan problems with nonlinear diffusion and convection

We consider a class of Stefan-type problems having a convection term and a pseudomonotone nonlinear diffusion operator. Assuming data in L¹, we prove existence, uniqueness and stability in the framework of renormalized solutions. Existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to L¹ convergence of the data. Uniqueness is proved through a classical L¹-contraction principle, obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators.     

Extrapolation for Classes of Weights Related to a Family of Operators and Applications

In this work we give extrapolation results on weighted Lebesgue spaces for weights associated to a family of operators. The starting point for the extrapolation can be the knowledge of boundedness on a particular Lebesgue space as well as the boundedness on the extremal case L ^∞. This analysis can be applied to a variety of operators appearing in the context of a Schrödinger operator (??Δ?+?V) where V satisfies a reverse Hölder inequality. In that case the weights involved are a localized version of Muckenhoupt weights.     

Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries

In this paper, we investigate the asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries. Under some suitable assumptions, we prove that the solution approaches a combination of Lipschitz continuous and piecewise C¹ traveling wave solution. As an application, we apply the result to the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n).     

Fujita phenomena in nonlinear pseudo-parabolic system

This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut-△u-αut=vp,vt-△v-αvt=uqwith p,q 1 and pq1,where the viscous terms of third order are included.We first find the critical Fujita exponent,and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions.Moreover,time-decay profiles are obtained for the global solutions.It can be found that,diferent from those for the situations of general semilinear heat systems,we have to use distinctive techniques to treat the influence from the viscous terms of the highest order.To fix the non-global solutions,we exploit the test function method,instead of the general Kaplan method for heat systems.To obtain the global solutions,we apply the Lp-Lq technique to establish some uniform Lmtime-decay estimates.In particular,under a suitable classification for the nonlinear parameters and the initial data,various Lmtime-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system.It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing efect to establish the compactness of approximating solutions,which cannot be directly realized here due to absence of the smooth efect in the pseudo-parabolic system.     

Boundedness of the extremal solution for some p-Laplacian problems

We investigate the regularity of extremal solutions to some p-Laplacian Dirichlet problems with strong nonlinearities. Under adequate assumptions we prove the smoothness of the extremal solutions for some classes of nonlinearities. Our results suggest that the extremal solution’s boundedness for some range of dimensions depends on the nonlinearity f.