On nonexistence of Baras-Goldstein type without positivity assumptions for singular linear and nonlinear parabolic equations

The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B ₁ ⊂ ℝ ^N , N ≥ 3, u _t = Δ _u + in B ₁ × (0,T), u|∂B ₁ = 0, in the supercritical range c > c _Hardy = does not have a solution for any nontrivial L ¹ initial data u ₀(x) ≥ 0 in B ₁ (or for a positive measure u ₀). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u _n (x,t)} of classical solutions corresponding to truncated bounded potentials given by V(x) = ↦ V _n (x) = min{, n} (n ≥ 1) diverges; i.e., as n → ∞, u _n (x,t) → + ∞ in B ₁ × (0, T). Similar features of “nonexistence via approximation” for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u ₀(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u _t = , and , N > 4. Dedicated to Professor S.I. Pohozaev on the occasion of his 70th birthday     

On a class of semilinear weakly hyperbolic equations

Abstract  In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

Numerical methods

In this work, we construct explicit travelling wave solutions involving parameters of the Drinfel’d–Sokolov–Wilson equation as

Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

We consider the following singularly perturbed boundary-value problem:

Supercritical biharmonic equations with power-type nonlinearity

We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation Δ² u = |u| ^p-1 u over the whole space , where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p _c, where p _c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case p ≥ p _c. We also study the Dirichlet problem for the equation Δ² u = λ (1 + u) ^p over the unit ball in , where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n + 4)/(n − 4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p _c. Finally, we show that a singular solution exists for some appropriate λ > 0.      

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      7. Fast and slow decay solutions for supercritical elliptic problems in exterior domains      Juan Dávila  Manuel del Pino  Monica Musso  Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2008,32(4):453-480 We consider the elliptic problem Δu  +  u p  =  0, u  >  0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded in , N ≥ 3, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay at infinity. In addition, a solution with fast decay O(|x|2-N ) exists if p is close enough from above to the critical exponent.      8. Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder      G. V. Grishina 《Mathematical Notes》1997,62(5):586-595 We study equations of the form $$\begin{gathered} u_{tt} + Lu + b(x,t)u_t = a(x,t)\left| u \right|^{\sigma - 1} u, \hfill \\ - u_t + Lu = a(x,t)\left| u \right|^{\sigma - 1} u \hfill \\ \end{gathered}$$ , whereL is a uniformly elliptic operator and 0<σ<1. In the half-cylinder II0,∞={(x, t):x= (x 1,...,x n )∈ ω,t>0}, where ? ? ? n is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition forx∈?ω andt>0. We find conditions under which these solutions have compact support and prove statements of the following type: ifu(x, t)=o(t γ) ast→∞, then there exists aT such thatu(x, t)≡0 fort>T. In this case γ depends on the coefficients of the equation and on the exponent σ.      9. Global solution and asymptotic behaviour of nonlinear Kirchhoff model in infinitely increasing moving domains      Rachid Benabidallah  François Ebobisse 《Annali dell'Universita di Ferrara》2001,47(1):207-229       10. Expansion of derivatives in one-dimensional dynamics      Bruin  Henk  van Strien  Sebastian 《Israel Journal of Mathematics》2003,137(1):223-263 We study the expansion of derivatives along orbits of real and complex one-dimensional mapsf, whose Julia setJ f attracts a finite setCrit of non-flat critical points. Assuming that for eachcεCrit, either |D f n(f(c))|→∞ (iff is real) orb n·|Df n(f(c))|→∞ for some summable sequence {b n} (iff is complex; this is equivalent to summability of |D f n(f(c))|−1), we show that for everyxεJ f\U i f −i(Crit), there existℓ(x)≤max c ℓ(c) andK′(x)>0 for infinitely manyn. Here 0=n s<…<n 1<n 0=n are so-called critical times,c i is a point inCrit (or a repelling periodic point in the boundary of the immediate basin of a hyperbolic periodic attractor), which shadows orb(x) forn i−ni +1 iterates, and, for uniform constantsK>0 and λ>1. If allcεCrit have the same critical order, thenK′(x) is uniformly bounded away from 0. Several corollaries are derived. In the complex case, eitherJ f= orJ f has zero Lebesgue measure. Also (assuming all critical points have the same order) there existk>0 such that ifn is the smallest integer such thatx enters a certain critical neighbourhood, then |Df n(x)|≥k. The original paper used an incorrect version of the Koebe Lemma cited from [21] as was pointed out by the referee and Genadi Levin in the autumn of 2001. The corrected version of November 2001 only uses the classical Koebe Lemma. Apparently, all results in Feliks Przytycki’s paper [21] go through using the classical Koebe Lemma instead of his Lemma 1.2. Both authors gratefully acknowledge the support of the PRODYN program of the European Science Foundation. HB was partially supported by a fellowship of The Royal Netherlands Academy of Arts and Sciences (KNAW). SvS was partially supported by GR/M82714/01.      11. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      12. On a class of nonlinear problems involving a p(x)-Laplace type operator      Mihai Mihăilescu 《Czechoslovak Mathematical Journal》2008,58(1):155-172 We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ N . Our attention is focused on two cases when , where m(x) = max{p 1(x), p 2(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.      13. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      14. A division problem for\bar \partial _b - CLOSED forms      Mats Andersson 《Journal d'Analyse Mathématique》1996,68(1):39-58 LetG 1,…,Gm be bounded holomorphic functions in a strictly pseudoconvex domainD such that . We prove that for each (0,q)-form ϕ inL p(∂D), 1<p<∞, there are formsu 1, …,u m inL p(∂D) such that ΣG juj=ϕ. This generalizes previous results forq=0. The proof consists in delicate estimates of integral representation formulas of solutions and relies on a certainT1 theorem due to Christ and Journé. For (0,n−1)-forms there is a simpler proof that also gives the result forp=∞. Restricted to one variable this is precisely the corona theorem. The author was partially supported by the Swedish Natural Research Council.      15. Growth of power series with square root gaps      Jacob Korevaar 《Journal d'Analyse Mathématique》2001,85(1):177-194 For entire functionsf whose power series have Hadamard gaps with ratio ≥1+α>1, Gaier has shown that the condition |f(x)|≤e x forx≥0 implies |f(z)|≤C αe|z| (*) for allz. Here the result is extended to the case of square root gaps, that is, , with , where α>0. Smaller gaps cannot work. In connection with his proof of the general high indices theorem for Borel summability, Gaier had shown that square root gaps imply . Having such an estimate, one can adapt Pitt’s Tauberian method for the restricted Borel high indices theorem to show that, in fact, , which implies (*). The author also states an equivalent distance formula involving monomialsx pke−xinL ∞ (0, ∞).      16. Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2      D. E. Edmunds  R. Kerman  J. Lang 《Journal d'Analyse Mathématique》2001,85(1):225-243 We consider the weighted Hardy integral operatorT:L 2(a, b) →L 2(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥p=(∫ a b |w(t)|p dt)1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.      17. Large solutions for equations involving the p-Laplacian and singular weights      Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,60(4):594-607 In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of , with u = +∞ on ∂Ω. Here is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.       18. Solutions to Some Open Problems in Fluid Dynamics      Linghai ZHANG 《数学年刊B辑(英文版)》2008,29(2):179-198 Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.      19. Construction of Rank 2 Semistable Torsion Free Sheaves Which Are Not Limit of Vector Bundles      Rosa M. Migo-Roig 《manuscripta mathematica》1993,80(1):89-94 We show the following theorem of compensated compactness type: Ifu n ⇁u weakly in the spaceH 1,p (Ω, ℝ k ) and if also in the sense of distributions then ∂α(∣∇u∣ p-2∂α u)=0. This result has applications in the partial regularity theory ofp-stationary mappings Ω→S k −1.      20. Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay      Filippo Gazzola  Hans-Christoph Grunau 《Calculus of Variations and Partial Differential Equations》2007,30(3):389-415 We are interested in stability/instability of the zero steady state of the superlinear parabolic equation u t + Δ2 u = |u| p-1 u in , where the exponent is considered in the “super-Fujita” range p > 1 + 4/n. We determine the corresponding limiting growth at infinity for the initial data giving rise to global bounded solutions. In the supercritical case p > (n + 4)/(n−4) this is related to the asymptotic behaviour of positive steady states, which the authors have recently studied. Moreover, it is shown that the solutions found for the parabolic problem decay to 0 at rate t −1/(p-1).

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Fast and slow decay solutions for supercritical elliptic problems in exterior domains

We consider the elliptic problem Δu  +  u ^p  =  0, u  >  0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded in , N ≥ 3, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay at infinity. In addition, a solution with fast decay O(|x|^2-N ) exists if p is close enough from above to the critical exponent.     

Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder

We study equations of the form $$\begin{gathered} u_{tt} + Lu + b(x,t)u_t = a(x,t)\left| u \right|^{\sigma - 1} u, \hfill \\ - u_t + Lu = a(x,t)\left| u \right|^{\sigma - 1} u \hfill \\ \end{gathered}$$ , whereL is a uniformly elliptic operator and 0<σ<1. In the half-cylinder II_0,∞={(x, t):x= (x ₁,...,x _n )∈ ω,t>0}, where ? ? ? _n is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition forx∈?ω andt>0. We find conditions under which these solutions have compact support and prove statements of the following type: ifu(x, t)=o(t ^γ) ast→∞, then there exists aT such thatu(x, t)≡0 fort>T. In this case γ depends on the coefficients of the equation and on the exponent σ.     

Expansion of derivatives in one-dimensional dynamics

We study the expansion of derivatives along orbits of real and complex one-dimensional mapsf, whose Julia setJ _f attracts a finite setCrit of non-flat critical points. Assuming that for eachcεCrit, either |D f ⁿ(f(c))|→∞ (iff is real) orb _n·|Df ⁿ(f(c))|→∞ for some summable sequence {b _n} (iff is complex; this is equivalent to summability of |D f ⁿ(f(c))|⁻¹), we show that for everyxεJ _f\U _i f ⁻ⁱ(Crit), there existℓ(x)≤max _c ℓ(c) andK′(x)>0

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

Global existence and uniqueness for the magnetic Hartree equation

In this paper, we consider

A division problem for\bar \partial _b - CLOSED forms

LetG ₁,…,G_m be bounded holomorphic functions in a strictly pseudoconvex domainD such that . We prove that for each (0,q)-form ϕ inL ^p(∂D), 1<p<∞, there are formsu ₁, …,u _m inL ^p(∂D) such that ΣG _ju_j=ϕ. This generalizes previous results forq=0. The proof consists in delicate estimates of integral representation formulas of solutions and relies on a certainT1 theorem due to Christ and Journé. For (0,n−1)-forms there is a simpler proof that also gives the result forp=∞. Restricted to one variable this is precisely the corona theorem. The author was partially supported by the Swedish Natural Research Council.     

Growth of power series with square root gaps

For entire functionsf whose power series have Hadamard gaps with ratio ≥1+α>1, Gaier has shown that the condition |f(x)|≤e ^x forx≥0 implies |f(z)|≤C _αe^|z| (*) for allz. Here the result is extended to the case of square root gaps, that is, , with , where α>0. Smaller gaps cannot work. In connection with his proof of the general high indices theorem for Borel summability, Gaier had shown that square root gaps imply . Having such an estimate, one can adapt Pitt’s Tauberian method for the restricted Borel high indices theorem to show that, in fact, , which implies (*). The author also states an equivalent distance formula involving monomialsx ^pke^−xinL ^∞ (0, ∞).     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

Large solutions for equations involving the p-Laplacian and singular weights

In this paper we consider the boundary blow-up problem Δ_pu = a(x)u^q in a smooth bounded domain Ω of , with u = +∞ on ∂Ω. Here is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.      

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

Construction of Rank 2 Semistable Torsion Free Sheaves Which Are Not Limit of Vector Bundles

We show the following theorem of compensated compactness type: Ifu _n ⇁u weakly in the spaceH ^1,p (Ω, ℝ ^k ) and if also in the sense of distributions then ∂_α(∣∇u∣ ^p-2∂_α u)=0. This result has applications in the partial regularity theory ofp-stationary mappings Ω→S ^k −1.     

Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay

We are interested in stability/instability of the zero steady state of the superlinear parabolic equation u _t + Δ² u = |u| ^p-1 u in , where the exponent is considered in the “super-Fujita” range p > 1 + 4/n. We determine the corresponding limiting growth at infinity for the initial data giving rise to global bounded solutions. In the supercritical case p > (n + 4)/(n−4) this is related to the asymptotic behaviour of positive steady states, which the authors have recently studied. Moreover, it is shown that the solutions found for the parabolic problem decay to 0 at rate t ^−1/(p-1).