共查询到20条相似文献,搜索用时 31 毫秒
1.
This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: ut+∇⋅(|∇Δu|p−2∇Δu)=f(u) in Ω⊂RN with boundary condition u=Δu=0 and initial data u0. The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u0 as p→∞. 相似文献
2.
We study extremal functions for a family of Poincaré-Sobolev-type inequalities. These functions minimize, for subcritical or critical p?2, the quotient ‖∇u‖2/‖up‖ among all u∈H1(B)?{0} with B∫u=0. Here B is the unit ball in RN. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of B. We also prove that, for p close to 2, minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis, and that, once the symmetry axis is fixed, they are unique (up to multiplication by a constant). In space dimension two, we prove that minimizers are not antisymmetric for large p. 相似文献
3.
Çi?dem A Bekta? 《Journal of Mathematical Analysis and Applications》2003,277(2):681-688
In this paper we define the sequence sets ?∞(u,Δ2,p), c(u,Δ2,p) and c0(u,Δ2,p), and give α- and β-duals of these sets. Further we investigate matrix transformations in the spaces and give a characterization of the class (?∞(u,Δ2,p),?∞). 相似文献
4.
Vitali Liskevich I.I. Skrypnik 《Journal of Mathematical Analysis and Applications》2008,338(1):536-544
We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a0(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|p−2+u|u|q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron. 相似文献
5.
Goro Akagi 《Journal of Differential Equations》2007,241(2):359-385
The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|p−2∇u), with initial data u0∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u0 and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T0] in which the problem admits a solution. More precisely, T0 depends only on Lr|u0| and f. 相似文献
6.
Bin Guo 《Journal of Mathematical Analysis and Applications》2011,374(2):374-384
The authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for ut=div((uσ+d0)|∇u|p(x,t)−2∇u)+f(x,t). Localization property of weak solutions is also discussed. 相似文献
7.
Zhijun Zhang Yiming Guo Huabing Feng 《Journal of Mathematical Analysis and Applications》2009,352(1):77-915
By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→0+g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary. 相似文献
8.
Bo Liang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(11):3815-3828
The paper first study the steady-state thin film type equation
∇⋅(un|∇Δu|q−2∇Δu)−δumΔu=f(x,u) 相似文献
9.
Mihai Mih?ilescu 《Journal of Mathematical Analysis and Applications》2007,330(1):416-432
We study the boundary value problem −div(log(1+q|∇u|)|∇u|p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|p−2u+|u|r−2u or f(u)=λ|u|p−2u−|u|r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces. 相似文献
10.
In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+up+εf(x), −Δu=μv+vq+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in RN (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ1) and ε,δ∈(0,δ0), where λ1 is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ0 is a positive number. 相似文献
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13.
Xiangsheng Xu 《Journal of Mathematical Analysis and Applications》2008,338(1):274-284
In this paper we present some regularity results for solutions to the system −Δu=σ(u)2|∇φ|, div(σ(u)∇φ)=0 in the case where σ(u) is allowed to oscillate between 0 and a positive number as u→∞. In particular, we show that u is locally bounded if σ(u) is bounded below by a suitable exponential function. 相似文献
14.
Tatiana V. Savina 《Journal of Mathematical Analysis and Applications》2010,370(2):716-1163
The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions u(x,y)∈R2 subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, Γ0:={y=0}, reflections are point-to-point when the given on Γ0 conditions are u=n∂u=0, u=Δu=0 or n∂u=n∂Δu=0, and point to a continuous set when u=n∂Δu=0 or n∂u=Δu=0 on Γ0. 相似文献
15.
The Cheeger problem for a bounded domain Ω⊂RN, N>1 consists in minimizing the quotients |∂E|/|E| among all smooth subdomains E⊂Ω and the Cheeger constant h(Ω) is the minimum of these quotients. Let be the p-torsion function, that is, the solution of torsional creep problem −Δp?p=1 in Ω, ?p=0 on ∂Ω, where Δpu:=div(|∇u|p−2∇u) is the p-Laplacian operator, p>1. The paper emphasizes the connection between these problems. We prove that . Moreover, we deduce the relation limp→1+‖?p‖L1(Ω)?CNlimp→1+‖?p‖L∞(Ω) where CN is a constant depending only of N and h(Ω), explicitely given in the paper. An eigenfunction u∈BV(Ω)∩L∞(Ω) of the Dirichlet 1-Laplacian is obtained as the strong L1 limit, as p→1+, of a subsequence of the family {?p/‖?p‖L1(Ω)}p>1. Almost all t-level sets Et of u are Cheeger sets and our estimates of u on the Cheeger set |E0| yield |B1|hN(B1)?|E0|hN(Ω), where B1 is the unit ball in RN. For Ω convex we obtain u=|E0|−1χE0. 相似文献
16.
In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu 0; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW 0 1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular. 相似文献
17.
We establish the critical Fujita exponents for the solution of the porous medium equation ut=Δum, x∈R+N, t>0, subject to the nonlinear boundary condition −∂um/∂x1=up, x1=0, t>0, in multi-dimension. 相似文献
18.
Zhijun Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(16):5544-5553
We study the existence, boundary behavior and uniqueness of solutions for the singular elliptic system −Δu=u−pv−q,−Δv=u−rv−s,u>0,v>0,x∈Ω,u|∂Ω=v|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, p,s≥0 and q,r>0. Our results are obtained in a range of p,q,r,s different from those in [M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal. 258 (2010) 3295-3318]. 相似文献
19.
Ahmed Hamydy 《Journal of Mathematical Analysis and Applications》2010,371(2):534-169
Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions. 相似文献
20.
Shingo Takeuchi 《Journal of Differential Equations》2011,251(8):2196-2208
This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|p−2∇u), 1<q?p<∞, f(s)∼|s|θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions. 相似文献