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1.
The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation with or in R
3×R
+ for small Cauchy data is proven if . A counterexample is given which shows that the lower bound on α is sharp.
Received April 1999 – Accepted September 1999 相似文献
2.
《Mathematical Methods in the Applied Sciences》2018,41(12):4506-4527
We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate Lq‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D1,r is the homogeneous Sobolev space. 相似文献
3.
Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity
Silvia Cingolani 《偏微分方程通讯》2013,38(8):1191-1203
ABSTRACT The existence of a nontrivial solution for quasi-linear elliptic equations involving the p-Laplace operator and a nonlinearity with p-linear growth at infinity is proved. Techniques of Morse theory are employed. 相似文献
4.
《随机分析与应用》2013,31(5):1189-1205
Abstract In this paper, we establish the existence of solutions of a more general class of stochastic functional integral equations. The main tools here are the measure of noncompactness and the fixed point theorem of Darbo type. The results of this paper generalize the results of Rao–Tsokos [Rao, A.N.V.; Tsokos, C.P. A class of stochastic functional integral equations. Coll. Math. 1976, 35, 141–146.] and Szynal–Wedrychowicz [Szynal, D.; Wedrychowicz, S. On existence and an asymptotic behaviour of random solutions of a class of stochastic functional integral equations. Coll. Math. 1987, 51, 349–364.]. 相似文献
5.
Alessio Figalli 《Journal of Functional Analysis》2008,254(1):109-153
In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the Stroock-Varadhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for Ld-almost every initial condition x whenever existence and uniqueness is known at the PDE level in the L∞-setting (and, conversely, if existence and uniqueness of martingale solutions is known for Ld-a.e. initial condition, then existence and uniqueness for the PDE holds). In the second part of the paper we consider situations where, on the one hand, no pointwise uniqueness result for the martingale problem is known and, on the other hand, well-posedness for the Fokker-Planck equation can be proved. Thus, the theory developed in the first part of the paper is applicable. In particular, we will study the Fokker-Planck equation in two somehow extreme situations: in the first one, assuming uniform ellipticity of the diffusion coefficients and Lipschitz regularity in time, we are able to prove existence and uniqueness in the L2-setting; in the second one we consider an additive noise and, assuming the drift b to have BV regularity and allowing the diffusion matrix a to be degenerate (also identically 0), we prove existence and uniqueness in the L∞-setting. Therefore, in these two situations, our theory yields existence, uniqueness and stability results for martingale solutions. 相似文献
6.
《Numerical Functional Analysis & Optimization》2013,34(2):229-247
ABSTRACT The existence of a countable set of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated by some variational methods based on the idea of the Fenchel conjugate. As a consequence of a duality developed here, we obtain the existence of a countable set of solutions for our problem that are minimizers to a certain integral functional. We derive (also in the superlinear case) a measure of a duality gap between primal and dual functional for approximate solutions. 相似文献
7.
Ying Sue Huang 《Journal of Difference Equations and Applications》2020,26(7):871-912
ABSTRACT We study a large class of finite difference equations that exhibit a type of periodic pattern repetition in their solutions for certain choices of initial conditions and prove the existence of unbounded solutions. 相似文献
8.
ABSTRACTIn this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results. 相似文献
9.
Mourad E. H. Ismail 《Journal of Difference Equations and Applications》2020,26(4):488-509
ABSTRACTWe introduce a finite difference and q-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or q-difference equation subject to a ‘terminating condition’, which is precisely defined. When a difference or q-difference equation has a polynomial solution, we show how to find the second solution. 相似文献
10.
We analyze the 2×2 nonhomogeneous relativistic Euler equations for perfect fluids in special relativity. We impose appropriate conditions on the lower order source terms and establish the existence of global entropy solutions of the Cauchy problem under these conditions. 相似文献
11.
Francisco de la Hoz 《Mathematische Zeitschrift》2007,257(1):61-80
In this paper, we will study self-similar solutions for X
t
= XẋX
ss
, the equivalent in the Minkowski 3-space to the localized induction approximation flow, trying to adapt some results given by Gutiérrez, Rivas and Vega. We will show
the existence of a one-parameter family of smooth solutions developing a corner in finite time. The main difference with respect
to the Euclidean case studied by those authors will be the proof of the boundedness of T, e
1 and e
2, the equivalents of T, b and n in .
The author was supported by the grant BFI02.135 of the Basque Government and by the project MTM2004-03029 of MEC (Spain) and
FEDER. 相似文献
12.
《随机分析与应用》2012,30(1):97-123
AbstractWe investigate the existence and uniqueness of strong solutions for state-dependent regime-switching diffusion processes in an infinite state space with singular coefficients. Non-explosion conditions are given by using the Zvonkin’s transformation. The strong Feller property is proved by further assuming that the diffusion in each fixed environment generates a strong Feller semigroup, and our results can also be applied to irregular or degenerate situations. 相似文献
13.
The existence of non-trivial solutions for nonlinear Dirichlet problems involving the p-Laplacian is investigated. In particular, an existence result of at least one non-trivial solution, without requiring any asymptotic condition on the nonlinear term either at zero or at infinity, is presented. As a consequence, also a multiplicity result is pointed out. The approach is based on a local minimum theorem for differentiable functionals. 相似文献
14.
Nikolaos Halidias 《随机分析与应用》2013,31(1):16-28
Abstract The purpose of this article is to consider a stochastic integral equation driven by semimartingale with discontinuous and increasing drift part. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered topological space. Finally we present some applications in finance. 相似文献
15.
《随机分析与应用》2013,31(5):1341-1361
Abstract In this paper we consider weak solutions to stochastic inclusions driven by a general semimartingale. We prove the existence of weak solutions and equivalence with the existence of solutions to the martingale problem formulated to such inclusion. Using this we then analyze compactness property of solutions set. Presenting results extend some of those being known for stochastic differential inclusions of Itô's type. 相似文献
16.
Piotr Pikuta Witold Rzymowski 《Journal of Mathematical Analysis and Applications》2003,277(1):122-129
In this paper we give a sufficient condition for the existence of maximal and minimal solutions to a discontinuous functional differential equation x′(t)=f(t,x(t),x(·)), x(0)=0. We apply the result to establish an existence theorem for the Darboux problem for a partial differential equation. 相似文献
17.
The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide. 相似文献
18.
《偏微分方程通讯》2013,38(1-2):121-138
Abstract In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions. 相似文献
19.
20.
《偏微分方程通讯》2013,38(1-2):409-438
Abstract We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem. 相似文献