We prove the nonexistence of solutions for a prescribed mean curvature equation when p?1 and the positive parameter λ is small. The result extends theorems of Narukawa and Suzuki, and Finn, from the case of n=2,p=1 to all n?2,p?1. Moreover, our proof is very simple and the result is not limited to positive (and negative) solutions. We also show that a similar result for positive solutions is still true if |u|p−1u is replaced by the exponential nonlinearity eu−1. 相似文献
In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3 with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space Wpl(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2). 相似文献
In this paper we condiser non-negative solutions of the initial value problem in ?N for the system where 0 ? δ ? 1 and pq > 0. We prove the following conditions. Suppose min(p,q)≥1 but pq1.
(a) If δ = 0 then u=v=0 is the only non-negative global solution of the system.
(b) If δ>0, non-negative non-globle solutions always exist for suitable initial values.
(c) If 0<?1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds.
(d) If N > 2, 0 < δ ? 1 and max (α β) < (N - 2)/2, then global, non-trivial non-negative solutions exist which belong to L∞(?N×[0, ∞]) and satisfy 0 < u(X, t) ? c∣x∣?2α and 0 < v(X, t) ? c ∣x∣?2bT for large ∣x∣ for all t > 0, where c depends only upon the initial data.
(e) Suppose 0 > δ 1 and max (α, β) < N/2. If N> = 1,2 or N > 2 and max (p, q)? N/(N-2), then global, non-trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H1 (K) where K(x) ? exp(¼∣x∣2). They decay like e[max(α,β)-(N/2)+ε]t for every ε > 0. These solutions are classical solutions for t > 0.
(f) If max (α, β) < N/2, then threre are global non-tivial solutions which satisfy, in the hot spot variables where where 0 < ε = ε(u0, v0) < (N/2)?;max(α, β). Suppose min(p, q) ? 1.
(g) If pq ≥ 1, all non-negative solutions are global. Suppose min(p, q) < 1.
(h) If pg > 1 and δ = 0, than all non-trivial non-negative maximal solutions are non-global.
(i) If 0 < δ ? 1, pq > 1 and max(α,β)≥ N/2 all non-trivial non-negative maximal solutions are non-global.
(j) If 0 < δ ≥ 1, pq > 1 and max(α,β) < N/2, there are both global and non-negative solutions.
We also indicate some extensions of these results to moe general systems and to othere geometries. 相似文献
We investigate the radially symmetric, nonlinear wave equation and discuss the asymptotic behaviour as r → ∞ of solutions which are T-periodic in time t. It is shown that only two possibilities of decay can arise, namely polynomial like t?½(n?1) and exponential e?bt for some b > 0. Existence results are obtained. 相似文献
We study the Cauchy problem for the quasilinear parabolic equation where p > 1 is a parameter and ψ is a smooth, bounded function on (1, ∞) with ? ? sψ′(s)/ψ(s) ? θ for some θ > 0. If 1 < p < 1 + 2/N, there are no global positive solutions, whereas if p > 1 + 2/N, there are global, positive solutions for small initial data. 相似文献
A basic mechanism of a formation of shocks via gradient blow‐up from analytic solutions for the third‐order nonlinear dispersion PDE from compacton theory (1) is studied. Various self‐similar solutions exhibiting single point gradient blow‐up in finite time, as t → T? < ∞ , with locally bounded final time profiles u(x, T?) , are constructed. These are shown to admit infinitely many discontinuous shock‐type similarity extensions for t > T , all of them satisfying generalized Rankine–Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow‐up is detected. This is in striking difference with general uniqueness‐entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics) (2) created by Oleinik in the middle of the 1950s. Contrary to (1) and not surprisingly, self‐similar gradient blow‐up for (2) is shown to admit a unique continuation. Bearing in mind the classic form (2) , the NDE (1) can be written as (3) with the standard linear integral operator (?D2x)?1 > 0 . However, because (3) is a nonlocal equation, no standard entropy and/or BV‐approaches apply (moreover, the x‐variations of solutions of (3) is increasing for BV data u0(x) ). 相似文献
Let ? be a convex function on a convex domain Ω⊂Rn, n?1. The corresponding linearized Monge–Ampère equation istrace(ΦD2u)=f, where is the matrix of cofactors of D2?. We establish interior Hölder estimates for derivatives of solutions to such equation when the function f on the right-hand side belongs to Lp(Ω) for some p>n. The function ? is assumed to be such that with ?=0 on ∂Ω and the Monge–Ampère measure is given by a density g∈C(Ω) which is bounded away from zero and infinity. 相似文献
We consider the Stokes system with resolvent parameter in an exterior domain: under Dirichlet boundary conditions. Here Ω is a bounded domain with C2 boundary, and [λ??\] ? [∞, 0], ν >0. Using the method of integral equations, we are able to construct solutions ( u , π) in Lp spaces. Our approach yields an integral representation of these solutions. By evaluating the corresponding integrals, we obtain Lp estimates that imply in particular that the Stokes operator on exterior domains generates an analytic semigroup in Lp. 相似文献
The Radon transform R(p, θ), θ∈Sn?1, p∈?1, of a compactly supported function f(x) with support in a ball Ba of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on Sn?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data. 相似文献