共查询到20条相似文献,搜索用时 31 毫秒
1.
We investigate the large-time behaviour of solutions to the nonlinear heat-conduction equation with absorption ut = Δ(uσ + 1) − uβ in Q = RN × (0, ∞) (E) with N 1, σ > 0 and critical absorption exponent β = σ + 1 + 2/N; the initial function u(x, 0) = 0 is assumed to be integrable, nonnegative and compactly supported. We prove that u converges as t → ∞ to a unique self-similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut = Δ(uσ + 1), the same for any initial data. The cornerstone of the proof is a result about ω-limits of (infinite-dimensional) asymptotical dynamical systems. Combining this result with an asymptotic evaluation of the mass function as well as typical PDE estimates gives the behaviour of (E) for large times.Similar unusual asymptotic behaviour is obtained for the equation ut = div(¦Du¦σ Du) − uβ with same conditions on σ and u(x, 0) and critical value for β = σ + 1 + (σ + 2)/N. 相似文献
2.
The nonlinear hyperbolic equation ∂2u(x, y)/∂x ∂y + g(x, y)f(u(x, y)) = 0 with u(x, 0) = φ(x) and u(0, y) = Ψ(y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986). 相似文献
3.
We consider the nonnegative solutions to the nonlinear degenerate parabolic equation ut = (D(x, t)um − 1ux)x − b(x, t)up with m > 1, 0 < p < 1, and positive D(x, t), b(x, t). After obtaining the uniqueness and Hölder regularity results, we investigate the dependence of such phenomena as extinction in finite time and instantaneous shrinking of the support on the behaviour of D(x, t) and b(x, t). 相似文献
4.
This paper deals with the Cauchy problem ut − uxx + up = 0; − ∞ < x < + ∞, t>0, u(x, 0) = u0(x); − ∞ < x < + ∞, where 0 < p < 1 and u0(x) is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time T, and interfaces separating the regions where u(x, t) > 0 and u(x, t) = 0 appear when t is close to T. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur. 相似文献
5.
B. Ricceri 《Mathematical and Computer Modelling》2000,32(11-13)
In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rn is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W01,2(Ω). 相似文献
6.
Fucai Li 《Applied Mathematics Letters》2004,17(12):686
This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationutt+(∫Ω׀Dmu׀2dx)q(−Δ)mu+ut׀ut׀r=׀u׀pu,xΩ,t>0,in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while ifp > max r, 2q , then for any initial data with negative initial energy, the solution blowsup at finite time in Lp+2 norm. 相似文献
7.
The wave equation for Dunkl operators 总被引:1,自引:0,他引:1
Let k = (kα)αε, be a positive-real valued multiplicity function related to a root system , and Δk be the Dunkl-Laplacian operator. For (x, t) ε N, × , denote by uk(x, t) the solution to the deformed wave equation Δkuk,(x, t) = δttuk(x, t), where the initial data belong to the Schwartz space on N. We prove that for k 0 and N l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N − 3)/2 + Σαε+kα ε . Here + is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of uk(x, t) is contained in the conical shell {(x, t), ε N × | |t| − R x |t| + R}. Our approach uses the representation theory of the group SL(2, ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of uk is, for large |t|, partitioned into equal potential and kinetic parts. 相似文献
8.
Let Ω be a bounded domain with smooth boundary in . For the more general weight b, some nonlinearities f and singularities g, by two kinds of nonlinear transformations, a new perturbation method, which was introduced by García Melián in [J. García Melián, Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal. 67 (2007) 818–826], and comparison principles, we show that the boundary behavior of solutions to a boundary blow-up elliptic problem Δw=b(x)f(w),w>0,xΩ,w|∂Ω=∞ and a singular Dirichlet problem −Δu=b(x)g(u),u>0,xΩ,u|∂Ω=0 has the same form under the nonlinear transformations, which can be determined in terms of the inverses of the transformations. 相似文献
9.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
10.
Xiaojing Yang 《Mathematische Nachrichten》2004,268(1):102-113
In this paper, the boundedness of all solutions of the nonlinear differential equation (φp(x′))′ + αφp(x+) – βφp(x–) + f(x) = e(t) is studied, where φp(u) = |u|p–2 u, p ≥ 2, α, β are positive constants such that = 2w–1 with w ∈ ?+\?, f is a bounded C5 function, e(t) ∈ C6 is 2πp‐periodic, x+ = max{x, 0}, x– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
11.
In this paper we study the existence of periodic solutions of the fourth-order equations uiv − pu″ − a(x)u + b(x)u3 = 0 and uiv − pu″ + a(x)u − b(x)u3 = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P1) and (P2) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P1) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P2) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation uiv + pu″ + a(x)u − b(x)u2 − c(x)u3 = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used. 相似文献
12.
Explosive solutions of elliptic equations with absorption and nonlinear gradient term 总被引:2,自引:0,他引:2
Marius Ghergu Constantin Niculescu Vicenţiu Rădulescu 《Proceedings Mathematical Sciences》2002,112(3):441-451
Letf be a non-decreasing C1-function such that
andF(t)/f
2
a(t)→ 0 ast → ∞, whereF(t)=∫
0
t
f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|
a
=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded. 相似文献
13.
Y. Guo 《Ukrainian Mathematical Journal》2011,62(9):1409-1419
We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations
*20c Da u(t) + l[ f( t,u(t) ) + q(t) ] = 0, 0 < t < 1, u(0) = 0, u(1) = bu(h), \begin{array}{*{20}{c}} {{{\mathbf{D}}^\alpha }u(t) + {{\lambda }}\left[ {f\left( {t,u(t)} \right) + q(t)} \right] = 0,\quad 0 < t < 1,} \\ {u(0) = 0,\quad u(1) = \beta u(\eta ),} \\ \end{array} 相似文献
14.
The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered:
u(4)(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0, 15.
On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary 总被引:1,自引:0,他引:1
We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu
t=uxx, ut=(um)xxand
(m>1) forx>0,t>0 with nonlinear boundary conditions−u
x=up,−(u
m)x=upand
forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove
that for each problem there exist positive critical valuesp
0,pc(withp
0<pc)such that forp∃(0,p
0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>p
csmall data solutions exist globally in time while large data solutions are nonglobal. We havep
c=2,p
c=m+1 andp
c=2m for each problem, whilep
0=1,p
0=1/2(m+1) andp
0=2m/(m+1) respectively.
This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications
at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210. 相似文献
16.
Guy Bernard 《Journal of Mathematical Analysis and Applications》1997,210(2):755
Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|p + f(x, t) = 0 in n × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation. 相似文献
17.
This paper deals with the existence of positive solutions for the nonlinear system
18.
In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k2+h2) for one, two and three space dimensional second-order hyperbolic equations utt=a(x,t)uxx+α(x,t)ux-2η2(x,t)u,utt=a(x,y,t)uxx+b(x,y,t)uyy+α(x,y,t)ux+β(x,y,t)uy-2η2(x,y,t)u, and utt=a(x,y,z,t)uxx+b(x,y,z,t)uyy+c(x,y,z,t)uzz+α(x,y,z,t)ux+β(x,y,z,t)uy+γ(x,y,z,t)uz-2η2(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k2) in order to obtain numerical solution of u at first time step in a different manner. 相似文献
19.
Joo-Paulo Dias Mrio Figueira Luis Sanchez 《Mathematical Methods in the Applied Sciences》1998,21(12):1107-1113
In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u0−(x) for x < x0, u(x, 0) = u0+(x) for x > x0, u0−(x0) > u0+(x0). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u0− and u0+, a global shock front weak solution u(x, t) = u−(x, t) for x < ϕ(t), u(x, t) = u+(x, t) for x > ϕ(t), where u− and u+ are the strong solutions corresponding (respectively) to u0− and u0+ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u−(ϕ(t), t) + u+(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd. 相似文献
20.
The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u + V(x)u =λf(x, u) in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic. 相似文献
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