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1.
Ping Zhang 《Applications of Mathematics》2006,51(4):427-466
In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and
some related problems. We first introduce the main tools, the L
p
Young measure theory and related compactness results, in the first section. Then we use the L
p
Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear
wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove
the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed.
In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic
equation, which is also the so-called vortex density equation arising from sup-conductivity. 相似文献
2.
Hao Wu 《Journal of Mathematical Analysis and Applications》2008,348(2):650-670
We consider a nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we use a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term f is real analytic. Moreover, we provide an estimate on the convergence rate. 相似文献
3.
We prove that for fields close enough to the first critical field, minimizers of the Ginzburg-Landau functional have a number
of vortices bounded independently from the Ginzburg-Landau parameter. This generalizes a result proved in [SS1] and shows
that locally minimizing solutions of the Ginzburg-Landau equation found in [S1, S3] are actually global minimizers. It also
gives a partial answer to a question raised by F. Bethuel and T. Rivière in [BR].
Received: 10 July 2002 / Accepted: 23 January 2002 / Published online: 5 September 2002 相似文献
4.
Pierre Rouchon 《Journal of Differential Equations》2003,193(1):75-94
We consider the nonlinear heat equation with nonlocal reaction term in space , in smoothly bounded domains. We prove the existence of a universal bound for all nonnegative global solutions of this equation. Moreover, in contrast with similar recent results for equations with local reaction terms, this is shown to hold for all p>1. As an interesting by-product of our proof, we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction. 相似文献
5.
In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity‐loss type, which causes the difficulty in high‐frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for n≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for n≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for n = 1. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
6.
Abdelkader Boucherif Ali S. Al-Qahtani Bilal Chanane 《Numerical Functional Analysis & Optimization》2013,34(6):730-747
We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions. 相似文献
7.
Removability of an Isolated Singularity of Solutions of Nonlinear Elliptic Equations with Absorption
I. I. Skrypnik 《Ukrainian Mathematical Journal》2005,57(7):1149-1166
We prove a priori estimates for singular solutions of nonlinear elliptic equations with absorption. Using these estimates, we establish precise
conditions for the behavior of the absorption term of the equation under which solutions with point singularities do not exist.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 972–988, July, 2005. 相似文献
8.
We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H2,2(?) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H2,2(?) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation. 相似文献
9.
Rafael Carreño-Bolaños Beatriz Juarez-Campos Pavel I. Naumkin 《Studies in Applied Mathematics》2020,145(1):137-149
We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy-type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions. 相似文献
10.
Philippe Souplet 《Annali di Matematica Pura ed Applicata》2002,181(4):427-436
We prove an a priori estimate and a universal bound for any global solution of the nonlinear degenerate reaction-diffusion
equation u
t
=Δu
m
+u
p
in a bounded domain with zero Dirichlet boundary conditions.
Received: October 1, 2001?Published online: July 9, 2002 相似文献
11.
We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel. 相似文献
12.
Raúl Ferreira Mayte Pérez‐Llanos 《Mathematical Methods in the Applied Sciences》2009,32(18):2439-2459
We study numerical approximations of solutions of the following system of heat equations, coupled at the boundary through a nonlinear flux: where p and q are parameters. We prove that the solutions of a semidiscretization in space quench in finite time. Moreover, we describe in terms of p and q the simultaneous versus non‐simultaneous quenching phenomena. We also find the numerical extinction sets. Finally, in order to obtain the correct quenching rates in the non‐simultaneous case we present some adaptive methods. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
13.
Nakao Hayashi Pavel I. Naumkin 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):6950-6964
We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions (0.1) We prove that if the initial data u0 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions. 相似文献
14.
Shuji Yoshikawa 《Mathematical Methods in the Applied Sciences》2005,28(12):1423-1443
We show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non‐linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23 : 299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
15.
Jérôme Droniou Alain Prignet 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(1-2):181-205
We consider the nonlinear heat equation (with Leray-Lions operators) on an open bounded subset of RN with Dirichlet homogeneous boundary conditions. The initial condition is in L1 and the right hand side is a smooth measure. We extend a previous notion of entropy solutions and prove that they coincide
with the renormalized solutions. 相似文献
16.
Linear and nonlinear heat equations in
$L^q_\delta$ spaces and universal bounds for global solutions
We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces , where is the distance to the boundary. In particular, we prove an optimal estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions
for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning
positive global solutions of superlinear parabolic problems. Namely, under certain assumptions, we prove that any global solution
is uniformly bounded for by a universal constant, independent of the initial data. In all previous results, the bounds for global solutions were depending
on the initial data.
Received March 15, 2000 / Accepted October 18, 2000 / Published online February 5, 2001 相似文献
17.
Sh. M. Nasibov 《Theoretical and Mathematical Physics》2018,195(2):658-664
We study the problem of the absence of global solutions of the first mixed problem for one nonlinear evolution equation of Schrödinger type. We prove that global solutions of the studied problem are absent for “sufficiently large” values of the initial data. 相似文献
18.
Hamid Ghidouche Philippe Souplet Domingo Tarzia 《Proceedings of the American Mathematical Society》2001,129(3):781-792
We consider a one-phase Stefan problem for the heat equation with a nonlinear reaction term. We first exhibit an energy condition, involving the initial data, under which the solution blows up in finite time in norm. We next prove that all global solutions are bounded and decay uniformly to 0, and that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial. Finally, we show that small data solutions behave like (i).
19.
Yong Hah Lee 《Mathematische Annalen》2000,318(1):181-204
We prove that if the s-harmonic boundary of a complete Riemannian manifold consists of finitely many points, then the set of bounded energy finite
solutions for certain nonlinear elliptic operators on the manifold is one to one corresponding to , where l is the cardinality of thes-harmonic boundary. We also prove that the finiteness of cardinality of s-harmonic boundary is a rough isometric invariant, moreover, in this case, the cardinality is preserved under rough isometries
between complete Riemannian manifolds. This result generalizes those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of
Kim and the present author, of Holopainen, and of the present author, but with different techniques which are demanded by
the peculiarity of nonlinearity.
Received October 13, 1999 / Revised November 23, 1999 / Published online July 20, 2000 相似文献