Large time behaviour of the solutions of a semilinear parabolic equation in RN

The asymptotic behaviour as t tends to +∞ of the solution of $\text{(?u}\text{?t)}\text{? Δu + u¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{= 0}$ in $\text{R}$^N × $\text{R}$⁺, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of $\text{(N + 2)}\text{N ? p}$ and also on the rate of decay of the admissible initial data u(0, x) as $\text{¦x¦}$ tends to +∞.     

Convergence of the Kato approximants for evolution equations involving functional perturbations

The existence of a unique strong solution of the nonlinear abstract functional differential equation $\text{u′(t) + A(t)u(t) = F(t,u}{}_{\mathrm{t}}\text{), u}{}_0\text{= φεC}{}^1\text{(¦?r,0¦,X),tε¦0, T¦}$, (E) is established. X is a Banach space with uniformly convex dual space and, for $\text{t? ¦0, T¦, A(t)}$ is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence u_n(t), where the functions u_n(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

Low energy scattering for nonlinear Klein-Gordon equations

The scattering operator which belongs to a pair of PDEs consisting of the Klein-Gordon equation and a perturbation of it by a power-like nonlinearity z.hfl;(u) is studied. It is shown that this operator can be defined on a whole neighbourhood of the origin in energy space if $\text{z.hfl;(u) = ±¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{u}\text{or}\text{±¦u¦}{}^{\mathrm{p}}$, where $\text{1+4}\text{(n ? 1)}\text{<p <}\text{1 + 4}\text{(n ? 2)}$ and the space dimension n ? 2 is arbitrary.     

On a property of the set of radiation patterns

Consider the exterior boundary value problem (▽² + K²) u = 0, in Ω, k >0. $\text{u¦}{}_{\mathrm{\Gamma }}\text{= h}$, where Γ is a smooth closed connected surface in $\text{R}$³, $\text{u ~}\text{exp}\text{(ik ¦x¦)¦x¦}{}^{\mathrm{?1}}\text{∝(k, n)}\text{as}\text{¦X¦→ ∞, n = x¦x¦}{}^{\mathrm{?1}}$, ∝ is called the radiation pattern. We prove that when h runs through any dense set in L²(Γ) the corresponding radiation pattern ∝(k,n) runs through a dense set in L²(S²) for any k >0, where S² is the unit sphere in $\text{R}$³.     

Über das grösstmögliche wachstum der nichtlinearität bei semilinearen parabolischen Gleichungen beliebiger ordnung

This paper deals with classical solvability for all t of semilinear parabolic equations u′ + A(t)u = f(t, x, u, ▽u, …, ▽^{2m ? 1}u). It is shown that the right side is allowed to grow faster than $\text{¦▽}{}^{\mathrm{m}}\text{u¦}{}^2$ in ▽^mu if a Hölder norm of u is known a priori. In the second part an example is given where an a priori estimate of a Hölder norm of u is available. Moreover, we give a new maximum principle.     

Sublinear perturbations of the differential equation y(n) = 0 and of the analogous difference equation

According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where $\text{y}{}^{\mathrm{(n)}}\text{(t)+}\text{∑}\text{i=0}\text{n?1}\text{g}{}_{\mathrm{i}}\text{(t) y}{}^{\mathrm{i}}\text{(t)=0}\text{(t ? 1)}$, $\text{∝}{}_1{}^{\mathrm{\infty }}\text{¦g}{}_{\mathrm{i}}\text{(x)¦x}{}^{\mathrm{n?I?1}}\text{dx < ∞}$ (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that $\text{lim}\text{t → ∞}\text{y(t)}\text{t}{}^{\mathrm{r}}$ exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials.     

Global existence for a delay differential equation

An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) $\text{min}{}_{\mathrm{<mtext>v</mtext><mtext>?</mtext><mtext>K</mtext>}}\text{{∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{v¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{v} = ∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{u¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{u, u}\text{?}\text{K,}\text{where}\text{(0.2) K = {v}\text{?}\text{H}{}_0{}^1\text{(Q), ¦}\text{▽}\text{v¦ ? 1}\text{a.e.}\text{}}\text{and}\text{θ}{}_1$ is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λ_θ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ₂ < θ₁. Let (0.4) $\text{K}{}^1\text{= {v}\text{?}\text{H}{}_0{}^1\text{(Q), ?d(x) ? v(x) ? d(x)},}\text{where}\text{d(x) =}\text{dist}\text{.(x, ?Q)}$, and denote by $\text{u}{}^1\text{, w}{}^1$ the solutions of (0.1), (0.3), respectively, when K is replaced by $\text{K}{}^1$. The following results are well known for the loading problem (0.1):(0.5) $\text{u = u}{}^1$; (0.6) the plastic set $\text{P = (X}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{u(x)¦ = 1}}$ is connected to the boundary. In this paper we show that, in general, (0.7) $\text{w ≠ w}{}^1$; (0.8) the plastic set $\text{P}\text{?}\text{= {x}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{w(x)¦ = 1}}$ is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ₁, θ₂.     

On Lp-spectrum of elliptic operators

For elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{a}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$ on Rⁿ and certain of their singular perturbations $\text{B = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x)D}{}^{\mathrm{\alpha }}$ relative compactness of B with respect to A is established. This result applies to the study of L^p-spectra of elliptic operators for different p.     

W1,p-quasiconvexity and variational problems for multiple integrals

Variational problems for the multiple integral $\text{I}{}_{\mathrm{\Omega }}\text{(u) = ∝}{}_{\mathrm{\Omega }}\text{g(▽u(x))dx}$, where $\text{Ω?R}{}^{\mathrm{m}}$ and $\text{u:Ω→R}{}^{\mathrm{n}}$ are studied. A new condition on g, called W^1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of $\text{I}{}_{\mathrm{\Omega }}$ in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$ and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$, p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.     

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

Radial bounds for perturbations of elliptic operators

Elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$, α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}$ or a quotient space $\text{R}{}^{\mathrm{n}}\text{R}{}^{\mathrm{n}}\text{U}{}_{\mathrm{\alpha }}\text{, U}{}_{\mathrm{\alpha }}\text{? R}{}^{\mathrm{n}}$ are considered. It is shown that the L^p-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L¹ radial convolution kernel. Some consequences are: A can be closed in all L^p (1 ? p ? ∞), and is essentially self-adjoint in L² if it is symmetric; A generates an analytic semigroup e^?tA in the right half plane, strongly L^p and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability $\text{φ(εA)?→}{}_{\mathrm{\epsilon \; \to \; 0}}\text{φ(0) ?}$, with φ analytic, is proved for $\text{? ? L}{}^{\mathrm{p}}$, with convergence in L^p and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients.     

Integral inequalities for generalized concave or convex functions

Let ψ be convex with respect to ?, B a convex body in Rⁿ and f a positive concave function on B. A well-known result by Berwald states that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{ψ(f(x)) dx ? n ∝}{}_0{}^1\text{ψ(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$ (1) if ξ is chosen such that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{?(f(x)) dx = n ∝}{}_0{}^1\text{?(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$.The main purpose in this paper is to characterize those functions f : B → R₊ such that (1) holds.     

Asymptotic behavior for an almost periodic,strongly dissipative wave equation

This paper deals with asymptotic behavior for (weak) solutions of the equation $\text{u}{}_{\mathrm{tt}}\text{? Δu + β(u}{}_{\mathrm{t}}\text{) ? ?(t, x)}$, on $\text{R}$⁺ × Ω; u(t, x) = 0, on $\text{R}$⁺ × ?Ω. If $\text{?∈L∞(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition $\text{?}{}_{\mathrm{t}}\text{∈S}{}^2\text{(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.     

Sharp nonlogarithmic Kneser theorems for fourth-order elliptic equations

Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim $\text{inf}\text{¦f(z) ¦ ? 1}$ on an arc A of ?Δ with length $\text{¦A ¦? ?}$. It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = sup_z?Δ$\text{(}\text{1 ? ¦z¦}{}^2\text{)¦f′(z)¦}\text{(1 + ¦f(z)¦}{}^2\text{) ? C}{}_1\text{sin}\text{(π ? (}\text{?}\text{2}\text{))/ (π ? (}\text{g9}\text{2}\text{))}$, where C₁ = lim_n→∞$\text{∥ z}{}^{\mathrm{n}}\text{∥}\text{and}\text{1}\text{2}\text{< C}{}_1\text{<}\text{2}\text{e}$. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that $\text{¦tf(e}{}^{\mathrm{i0}}\text{)¦ = 1}$ almost everywhere. It is proved that inf_f?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C₁. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥f^p∥ ? p∥f∥, for any positive integer p.     

A regularity result for singular nonlinear elliptic systems in inverse-power weighted Sobolev spaces

The compactness method to weighted spaces is extended to prove the following theorem:Let H_2,s¹(B₁) be the weighted Sobolev space on the unit ball in Rⁿ with norm

$\text{6ν6}{}^1{}_{\mathrm{2,s}}\text{=}\text{∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|ν|}{}^2\text{dx + ∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|Dν|}{}^2\text{dx.}$

Let n ? 2 ? s < n. Let u? [H_2,s¹(B₁) ∩ L^∞(B₁)]^N be a solution of the nonlinear elliptic system

$\text{∫}{}_{\mathrm{B1}}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{i,j=1}\text{n}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{(x,u) D}{}^{\mathrm{i}}\text{u}{}_{\mathrm{h}}\text{D}{}^{\mathrm{j\psi }}{}_{\mathrm{K}}\text{dx=0}$

, $\text{∨}\text{ψ}\text{?}\text{¦C}{}_0{}^1\text{(B}{}_1\text{)¦}{}^{\mathrm{N}}\text{,}\text{where}\text{¦A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}\text{¦ ? L, A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}$ are uniformly continuous functions of their arguments and satisfy:

$\text{|η|}{}^2\text{=}\text{∑}\text{i=1}\text{n}\text{,}\text{∑}\text{j=1}\text{N}\text{|η}{}^{\mathrm{i}}{}_{\mathrm{j}}\text{|}{}^2\text{?}\text{∑}\text{i,j=1}\text{n}\text{,}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{h}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{k}}\text{,}\text{?η?R}{}^{\mathrm{Nn}}$

. Then there exists an R₁, 0 < R₁ < 1, and an α, 0 < α < 1, along with a set $\text{Ω ? B}{}_1$ such that (1) $\text{H}{}_{\mathrm{n\; ?\; 2}}\text{(Ω) = 0}$, (2) Ω does not contain the origin; Ω does not contain B_R1, (3) $\text{B}{}_1\text{? Ω}$ is open, (4) u is $\text{Lip}{}_{\mathrm{\alpha }}\text{(B}{}_1\text{? Ω)}$; u is Lip_αB_R1.     

Equations du type de Monge-Ampère sur les variétés Riemanniennes compactes,III

Let (V_n, g) be a C^∞ compact Riemannian manifold without boundary. Given the following changes of metric: $\text{g′}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{= g +}\text{Hess}\text{? ±}\text{l}\text{α}{}^2\text{(▽ ? ? ▽?),}\text{g}\text{?}{}^{\mathrm{\pm }}\text{= ±?g + α}{}^2\text{Hess}\text{?}$, where a is a fixed constant, we study the corresponding Monge-Ampère equations (1)^±$\text{Log}\text{(¦g′}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{¦ ¦g¦}{}^{\mathrm{?1}}\text{) = F(P,▽?;?)}$, (2)^±$\text{Log}\text{(¦}\text{g}\text{?}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{¦ ¦g¦}{}^{\mathrm{?1}}\text{) = F(P, ▽?; ?)}$. We first solve Eq. (2)^?, under some simple assumptions on F?C^∞. Then, using an appropriate change of functions that enables us to take advantage of the estimates just carried out for Eq. (2)^?, we extend to Eq.(1)^? all the results proved in our previous articles [5, 6] for the usual Monge-Ampère equation. Although equation (2)⁺ is not locally invertible, and does not even admit a solution for all $\text{F = λ? + ?, λ > 0, f ? C}{}^{\mathrm{\infty }}\text{(V}{}_{\mathrm{n}}\text{)}$, a similar change of functions leads to partial results about Eq. (1)⁺, via C² and C³ estimates for Eq. (2)⁺. Eventually we give some comments and errata of our previous article (P. Delanoë, J. Funct. Anal.41 (1981), 341–353).