共查询到20条相似文献,搜索用时 31 毫秒
1.
This paper studies heat equation with variable exponent u t = Δu + up(x) + u q in ? N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p? = inf p(x) ≤ p(x) ≤ sup p(x) = p+. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p+, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p+ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p? ≤ p+ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p? < 1 + \(\frac{2}{N}\) < p+. 相似文献
2.
A. M. Bikchentaev 《Mathematical Notes》2016,100(3-4):515-525
Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let Lp(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ Lp(M, τ), then A2 ∈ Lp(M, τ). Let n be a positive integer, n > 2, and A = An ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μt(A) belong to the set {0} ∪ [‖An?2‖?1, ‖A‖] for all t > 0; 2) either μt(A) ≥ 1 for all t > 0 or there is a t0 > 0 such that μt(A) = 0 for all t > t0. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z2 = 0, ZP = 0, and PZ = Z. Here, if Q ∈ Lp(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L1(M, τ) and if A = A3 and A ? A2 ∈ M, then τ(A) ∈ R. 相似文献
3.
We discuss the discrete p-Laplacian eigenvalue problem, where T > 1 is a given positive integer and φ p (x):= |x| p?2 x, p > 1. First, the existence of an unbounded continuum C of positive solutions emanating from (λ, u) = (0, 0) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any λ > 0 and all solutions are ordered. Thus the continuum C is a monotone continuous curve globally defined for all λ > 0.
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$$\left\{ \begin{gathered} \Delta (\phi _p (\Delta u(k - 1))) + \lambda a(k)g(u(k)) = 0,k \in \{ 1,2,...,T\} , \hfill \\ u(0) = u(T + 1) = 0, \hfill \\ \end{gathered} \right.$$
4.
Let G = (V,E) be a finite connected weighted graph, and assume 1 ? α ? p ? q. In this paper, we consider the p-th Yamabe type equation ―?pu+huq―1 = λfuα―1 on G, where ?p is the p-th discrete graph Laplacian, h < 0 and f > 0 are real functions defined on all vertices of G. Instead of H. Ge’s approach [Proc. Amer. Math. Soc., 2018, 146(5): 2219–2224], we adopt a new approach, and prove that the above equation always has a positive solution u > 0 for some constant λ ∈ ?. In particular, when q = p, our result generalizes Ge’s main theorem from the case of α ? p > 1 to the case of 1 ? α ? p, It is interesting that our new approach can also work in the case of α ? p > 1. 相似文献
5.
We study a nonlinear equation in the half-space {x 1 > 0} with a Hardy potential, specifically where p > 1 and ?∞ < μ < 1/4. The admissible boundary behavior of the positive solutions is either O(x 1 ?2/(p?1)) as x 1 → 0, or is determined by the solutions of the linear problem \( - \Delta h - \frac{\mu }{{x_1^2}}h = 0\). In the first part we study in full detail the separable solutions of the linear equations for the whole range of μ. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like O(x 1 ?2/(p?1)) near the origin and which, away from the origin, have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like O(x 1 ?2/(p?1)) at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.
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$$ - \Delta u - \frac{\mu }{{x_1^2}}u + {u^p} = 0in\mathbb{R}_ + ^n,$$
6.
A. S. Makin 《Differential Equations》2017,53(6):719-724
We obtain asymptotic formulas uniform with respect to the index p > 0 for the Hankel functions H p (j) (z) (j = 1, 2) for large |z| in the complex domain. These formulas generalize those known for the real argument. 相似文献
7.
A. S. Makin 《Differential Equations》2017,53(5):595-606
We obtain asymptotic formulas uniform with respect to the index p > 0 for the Hankel functions H p (j)(z) (j = 1, 2) for large |z| in the complex domain. These formulas generalize those well known for the real argument. 相似文献
8.
We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type ?Δ p u = a(x)u m ?b(x)f(u) with p > 1 and 0 < m < p?1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p = 2 to the quasilinear case with p > 1. 相似文献
9.
This paper studies the weighted, fractional Bernstein inequality for spherical polynomials on Sd-1\(\left( {0.1} \right)\;{\left\| {{{\left( { - {\Delta _0}} \right)}^{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}}f} \right\|_{p,w}} \leqslant {C_w}{n^r}{\left\| f \right\|_{p,w}}\;for\;all\;f \in \Pi _n^d\), where Πnd denotes the space of all spherical polynomials of degree at most n on Sd-1 and (-Δ0)r/2 is the fractional Laplacian-Beltrami operator on Sd-1. A new class of doubling weights with conditions weaker than the Ap condition is introduced and used to characterize completely those doubling weights w on Sd-1 for which the weighted Bernstein inequality (0.1) holds for some 1 ≤ p ≤ 8 and all r > t. It is shown that in the unweighted case, if 0 < p < 8 and r > 0 is not an even integer, (0.1) with w = 1 holds if and only if r > (d - 1)((1/p) - 1). As applications, we show that every function f ∈ Lp(Sd-1) with 0 < p < 1 can be approximated by the de la Vallée Poussin means of a Fourier-Laplace series and establish a sharp Sobolev type embedding theorem for the weighted Besov spaces with respect to general doubling weights. 相似文献
10.
Liouville type results for a <Emphasis Type="Italic">p</Emphasis>-Laplace equation with negative exponent
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Positive entire solutions of the equation \(\Delta _p u = u^{ - q} in \mathbb{R}^N (N \geqslant 2)\) where 1 < p ≤ N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q c such that this equation has no positive radial entire solution that has finite Morse index when q > q c but it admits a family of stable positive radial entire solutions when 0 < q ≤ q c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q ≤ q c relies on Caffarelli–Kohn–Nirenberg’s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p ≤ N and q > q c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator. 相似文献
11.
Results on the convergence of minimizers and minimum values of integral and more general functionals Js: W1,p(Ωs) → ? on the sets Us(hs) = {v ∈ W1,p(Ωs): hs(v) ≤ 0 a.e. in Ωs}, where p > 1, {Ωs} is a sequence of domains contained in a bounded domain Ω of ?n (n > 2), and {hs} is a sequence of functions on ?, are announced. 相似文献
12.
Mei-Chu Chang 《Israel Journal of Mathematics》2017,222(2):631-643
Let G be a multiplicative subgroup of the prime field F p of size |G| > p1?κ and r an arbitrarily fixed positive integer. Assuming κ = κ(r) > 0 and p large enough, it is shown that any proportional subset A ? G contains non-trivial arithmetic progressions of length r. The main ingredient is the Szemerédi–Green–Tao theorem. 相似文献
13.
Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E2-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E2-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p2 + (s + 4)p3 + pn + pm]. 相似文献
14.
We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power where Ω is either a bounded domain or the whole space ? N , q(x) is a positive and continuous function defined in Ω with 0 < q ? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q + < p ? 1, all the solutions are global. If q ? > p ? 1, there exist global solutions, while for given q ? < p ? 1 < q +, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? N , the Fujita phenomenon occurs if 1 < q ? ? q + ? p ? 1 + p/N, while if q ? > p ? 1 + p/N, there exist global solutions.
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$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$
15.
This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?p(x′))′ + b(t)?p(x′) + c(t)?p(x) = 0, where ?p(x) = |x|p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented. 相似文献
16.
V. V. Volkov 《Vestnik St. Petersburg University: Mathematics》2016,49(4):320-324
In a two-dimensional local field K containing the pth root of unity, a polynomial formal group F c (X, Y) = X + Y + cXY acting on the maximal ideal M of the ring of integers б K and a constructive Hilbert pairing {·, ·} c : K 2(K) × F c (M) → <ξ> c , where <ξ> c is the module of roots of [p] c (pth degree isogeny of F c ) with respect to formal summation are considered. For the extension of two-dimensional local fields L/K, a norm map of Milnor groups Norm: K 2(L) → K 2(K) is considered. Its images are called norms in K 2(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β} c : = 0 ? x is a norm in K 2(K([p] c -1 (β))), where [p] c -1 (β) are the roots of the equation [p] c = β, is checked constructively. 相似文献
17.
Jiecheng Chen Belay Mitiku Damtew Xiangrong Zhu 《Frontiers of Mathematics in China》2017,12(2):281-299
We consider the oscillatory hyper Hilbert transform H γ,α,β f(x) = ∫ 0 ∞ f(x - Γ(t))eit-β t-(1+α)dt; where Γ(t) = (t, γ(t)) in ?2 is a general curve. When γ is convex, we give a simple condition on γ such that H γ,α,β is bounded on L 2 when β > 3α, β > 0: As a corollary, under this condition, we obtain the L p -boundedness of H γ,α,β when 2β/(2β - 3α) < p < 2β/(3α). When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H γ,α,β is bounded on L 2: As an application, we construct a class of strictly convex curves along which H γ,α,β is bounded on L 2 only if β > 2α > 0. 相似文献
18.
Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p 0, p 1 ∈ (1,∞] be such that (1 ? θ)/p 0 + θ/p 1 = 1/p, and let φ, φ0, φ1 be some admissible functions such that φ, φ0 p/p0 and φ1 p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L φ0 p0),λ (X), L φ1 p1), λ (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L φ p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces. 相似文献
19.
D. I. Tolev 《Proceedings of the Steklov Institute of Mathematics》2017,299(1):246-267
We consider the Diophantine inequality |p 1 c + p 2 c + p 3 c ? N| < (logN)?E, where 1 < c < 15/14, N is a sufficiently large real number and E > 0 is an arbitrarily large constant. We prove that the above inequality has a solution in primes p1, p2, p3 such that each of the numbers p1 + 2, p2 + 2 and p3 + 2 has at most [369/(180 ? 168c)] prime factors, counted with multiplicity. 相似文献
20.
V. M. Badkov 《Proceedings of the Steklov Institute of Mathematics》2009,264(1):39-43
Let {p n (t)} n=0 t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x n,ν (p) } ν=1 n be zeros of a polynomial p n (t) (x x,ν (p) = cosθ n,ν (p) ; 0 < θ n,1 (p) < θ n,2 (p) < ... < θ n,n (p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ n,1 (p) and 1 ? x n,1 (p) coincide in order with n ?1 and n ?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t 2)?1{ln2[(1 + t)/(1 ? t)] + π 2}?1, then the following asymptotic formulas are valid as n → ∞:
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$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$