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1.
A quasilinear equation u -x·u/2+f(u)=0 is studied, wheref(u)=–u+u , > 0, 0<. <1, >1 andx R n. The equation arises from the study of blow-up self-similar solutions of the heat equation t =+. We prove the existence and non-existence of ground state for various combination of , and . In particular, we prove that when / < forn=1,2 or / < (n + 2) /(n – 2) forn 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where / > (n + 2)/(n – 2) there exists an infinite number of non-constant positive radial self-similar solutions.  相似文献   

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In this article,a degenerate and singular diffusion problem is studied.The ex- istence of solutions is established by parabolic regularization.Some properties of solutions, for instance,asymptotic behavior,are also discussed.  相似文献   

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In this paper, we study discontinuous solutions of a partial differential equation of strongly degenerate parabolic type. A notion of weak solutions of BV class is proposed, and existence and uniqueness results are obtained.  相似文献   

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We present a systematic study of local solutions of the ODE of the form near t=0. Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ>0 and γ<0, where γ=∂f/∂x at t=0. As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.  相似文献   

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This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L2 decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L decay estimate is derived.  相似文献   

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Let a1,a2, . . . ,am ∈ ℝ2, 2≤fC([0,∞)), giC([0,∞)) be such that 0≤gi(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|xai|→−gi(t) as |xai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain.  相似文献   

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This paper is concerned with the Hölder estimates of weak solutions of the Cauchy problem for the general degenerate parabolic equations


with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.

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In this paper we study the shrinking self-similar solutions of the nonlinear diffusion equation with nondivergence form
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Existence, uniqueness, and non-uniqueness results are given for a class of singular diffusion problems with variable diffusion rates. Continuity arguments are used on related initial value problems to produce the existence results. Uniqueness criteria are established for different sets of hypotheses and non-uniqueness results are discussed when these hypotheses fail.  相似文献   

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 We prove that the solution u of the equation u t =Δlog u, u>0, in (Ω\{x 0})×(0,T), Ω⊂ℝ2, has removable singularities at {x 0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C 1, C 2>0, such that C 1 |xx 0|αu(x,t)≤C 2|xx 0|−α holds for all 0<|xx 0|≤ρ0 and atb. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u 0L 1 (ℝ2) is radially symmetric and u 0L loc 1(ℝ2). Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003 Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65  相似文献   

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Convergence to self-similar profiles is shown for solutions to the Oort-Hulst-Safronov coagulation equation with constant coagulation kernel. A dynamical systems approach is used on the equation written in self-similar variables, for which two Liapunov functionals are identified. For initial data decaying sufficiently rapidly at infinity, decay rates are also obtained.Received: October 17, 2002  相似文献   

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1.IntroductionInthispaper,weareconcernedwiththebehaviorofsolutionsofthefollowingproblemwheretheinitialdata"o(x)satisfiesnoEW"oo(a,b)and"o(x)20a.e.in(a,6).(1.4)Definition1.AnonnegativefunctionuELoo([0,co);W'loo(n))withfi=(a,b)iscalledaweaksolutionoftheproblem(1.1)--(1.3)ifthefollowingconditionsarefulfilled:(i)acEL'(flx[0,F])foranyT>0,(n)usatisfiestheX..d.yvaluecondition(l.2)intheusualsense.(iii)FOranytestfunctiongbEC2)1(fix[0, co))withgb(a,t)~op(b,t)~0,thereholdstheintegralequalityRecellt…  相似文献   

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In this paper, a singular approach to study the solutions of an impulsive differential equation from a qualitative and quantitative point of view is proposed. In the approach, a suitable singular perturbation term is introduced and a singularly perturbed system with infinite initial values is defined, in which, the reduced problem of the singularly perturbed system is exactly the impulsive differential equation under consideration. Then the boundary layer function method is applied to construct the uniformly valid asymptotic solutions to the singularly perturbed system. Based on the continuous asymptotic solution, the discontinuous solutions of the impulsive differential equation are described and approximated. An example, namely, a classical Lotka-Volterra prey-predator model with one pulse is carried out to illustrate the main results.  相似文献   

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