Interior numerical approximation of boundary value problems with a distributional data

We study the approximation properties of a harmonic function u ∈ H^1?k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = ??, for a smooth, bounded domain ??, we obtain that the GFEM‐approximation u_S ∈ S of u satisfies ‖u ? u_S‖ ≤ Ch^γ‖u‖, where h is the typical size of the “elements” defining the GFEM‐space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super‐approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161–168; Math Comput 28 (1974), 937–958). In addition to the usual “energy” Sobolev spaces H¹(??), we need also the duals of the Sobolev spaces H^m(??), m ∈ ?₊. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

Existence results for non-autonomous multiple-fragmentation models

We investigate an initial-value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non-negative, mass-conserving solutions is established by considering a truncated version of an associated non-autonomous abstract Cauchy problem. The latter has solutions of the form u(t)=U_n(t,t₀)f, t⩾t₀, where f is the known data at some fixed time t₀⩾0 and {U_n(t,s)} is a uniformly continuous evolution system. A limit evolution system {U(t,s)} is shown to exist. Depending on the form of the known data f at time t₀, the scalar-valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t₀)f](x) for a.e. x>0, t⩾t₀, is a solution of either the original initial-value problem or an integral version of this problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Strong solutions to the Keller‐Segel system with the weak Ln /2 initial data and its application to the blow‐up rate

We shall show an exact time interval for the existence of local strong solutions to the Keller‐Segel system with the initial data u₀ in L^{n /2}_w (?ⁿ), the weak L^{n /2}‐space on ?ⁿ. If ‖u₀‖ is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L^{n /2}_w (?ⁿ) stems from obtaining a self‐similar solution which does not belong to any usual L^p(?ⁿ). Furthermore, the characterization of local existence of solutions gives us an explicit blow‐up rate of ‖u (t)‖ for n /2 < p < ∞ as t → T_max, where T_max denotes the maximal existence time (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stability result for abstract evolution problems

Consider an abstract evolution problem in a Hilbert space H (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖^p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half‐plane Re(λ) < λ₀(t), λ₀(t) > 0, but assume that lim_{t → ∞}λ₀(t) = 0. Under suitable assumptions on a(t) and b(t), the boundedness of ‖u(t)‖ as t → ∞ is proved. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u₀ = 0 is established. For f ≠ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half‐plane Re(λ) < λ₀(t) with λ₀(t) > 0 and lim_{t → ∞}λ₀(t) = 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem (1) where q ∈ C⁰([0,1]) and f, h ∈ C⁰((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution x ∈ C¹([0,1]) ∩ C²((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.     

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i? _t u + Δu + |u|² u = 0 such that ‖xu(t)‖ _{L ²}  = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|² Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u ₀‖ _{L ²} ‖?u ₀‖ _{L ²}  > ‖Q‖ _{L ²} ‖?Q‖ _{L ²} , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t _n  → + ∞ such that ‖?u(t _n )‖ _{L ²}  → ∞. Similar statements hold for negative time.     

Certain classes of potentials for p ‐Laplacian to be non‐degenerate

Given a positive integer n and an exponent 1 ≤ α ≤ ∞. We will find explicitly the optimal bound r_n such that if the L^α norm of a potential q (t ) satisfies ‖q ‖equation/tex2gif-inf-2.gif < r_n then the n ^th Dirichlet eigenvalue of the onedimensional p ‐Laplacian with the potential q (t ): (|u ′|^{p –2} u ′)′ + (λ + q (t )) |u |^{p –2}u = 0 (1 < p < ∞) will be positive. Using these bounds, we will construct, for the Dirichlet, the Neumann, the periodic or the antiperiodic boundary conditions, certain classes of potentials q (t ) so that the p ‐Laplacian with the potential q (t ) is non‐degenerate, which means that the above equation with λ = 0 has only the trivial solution verifying the corresponding boundary condition. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫ |∇u|² : ∇u ∈ L²(R₊ⁿ), ∫ |u|^q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as (M, g), that there exists some constant A = A(M, g) > 0, (∫_dM|u|^q ds_g)^2/q < or = to S ∫_M |∇_gu|² dv_g + A ∫_dMu² ds_g, for all u ∈ H¹ (M). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc.     

Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.     

Decay estimates for the wave and Dirac equations with a magnetic potential

We study the electromagnetic wave equation and the perturbed massless Dirac equation on ℝ_t × ℝ³: where the potentials A(x), B(x), and V(x) are assumed to be small but may be rough. For both equations, we prove the expected time decay rate of the solution where the norm ‖f‖_X can be expressed as the weighted L²-norm of a few derivatives of the data f. © 2006 Wiley Periodicals, Inc.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

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 ⊂R^N, 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.     

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

A Threshold Result for a Non-local Parabolic Equation

In this paper, we consider the Cauchy problem: (ECP) u_t−Δu+p(x)u=u(x,t)∫u²(y,t)/∣x−y∣dy; x∈ℝ³, t>0, u(x, 0)=u₀(x)⩾0 x∈ℝ³, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ³, p(x)⩾ā>0, and lim_{∣x∣→∞}p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u₀(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u₀(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

A basic inequality for the Stokes operator related to the Navier boundary condition

We show that ‖Au+Δu_‖L²(Ωε)?C(ε‖∇u_‖L²(Ωε)+‖u_‖L²(Ωε)), where Ωε is a thin domain in R³ of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition.     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of boundary optimal controls in mixed elliptic problems

We consider a steady-state heat conduction problem P_α withmixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ₁ of the boundary of Ω. We consider, for each α > 0, a cost function J_α and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ₂ of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ₂ in terms of the adjoint state. We prove that the optimal control q and its corresponding system state u and adjoint state p for each α are strongly convergent to q_op, u and p in L²(Γ₂), H¹(Ω), and H¹(Ω) respectively when α → ∞. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition on the portion Γ₁. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux q defined on Γ₂ and the parameter α (which goes to infinity) is defined on Γ₁. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)