Lp Decay problem for the dissipative wave equation in even dimensions

We study L^p decay estimates of the solution to the Cauchy problem for the dissipative wave equation in even dimensions: (□+?_t)u=0 in ?^N × (0,∞) for even N=2n?2 with initial data (u,?_tu)∣_t=0 =(u₀,u₁). The representation formulas of the solution u(t)=?_tS(t)u₀ + S(t)(u₀+u₁) provide the sharp estimates on L^p norms with p?1. Copyright © 2004 John Wiley & Sons, Ltd.     

Approximate solution of u′=–A2u using a relation between exp(–tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u _t =–A² u. Using a representation of the semi group exp(–A² t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w _t = w _yy , with initial values v solving the initial value problem for v _y = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2^nd order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4th order equation for u to that of the 2^nd order equation for v, followed by the solution of the heat equation in one space variable.     

decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂_t)u=0 in with (u,∂_tu)|_t=0=(u₀,u₁). Because the L² estimates and the L^∞ estimates of the solution u(t) are well known, in this paper we pay attention to the L^p estimates with 1p<2 (in particular, p=1) of the solution u(t) for t0. In order to derive L^p estimates we first give the representation formulas of the solution u(t)=∂_tS(t)u₀+S(t)(u₀+u₁) and then we directly estimate the exact solution S(t)g and its derivative ∂_tS(t)g of the dissipative wave equation with the initial data (u₀,u₁)=(0,g). In particular, when p=1 and n1, we get the L¹ estimate: u(t)_L¹Ce^−t/4(u_0W^n,1+u_1W^n−1,1)+C(u_0L¹+u_1L¹) for t0.     

Convergence of the method of characteristics in approximate solution of a semilinear partial differential equation of first order

We construct an approximate solution for an initial boundary-value problem of the formu _t (x, t) + a (x, t) u_x (x, t)=b (x, t, u), u (x, 0)=u₀ (x),u (0,t)=u₁ (t) by the method of characteristics. It is proved that the approximate solution converges to the exact one with rate of convergence of second order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1128–1138, August, 1990.     

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

The existence of ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by with the initial condition u|_(t=0)=u₀, u_t|_(t=0)=u₁, and θ|_(t=0)=θ₀ in Ω and the boundary condition u=∂_νu=θ=0 on Γ, where u=u(x,t) denotes a vertical displacement at time t at the point x=(x₁,⋯,x_n)∈Ω, while θ=θ(x,t) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C₀ analytic semigroup and the maximal L_p‐L_q‐regularity of time‐dependent problem are derived.     

Consistent interpolation of equidistantly sampled data

Let u₁, u₂, …, u_N with u_n∈? denote the values of a function recorded or computed at N real and equidistant abscissa values t_n=nΔt+t₀ for n=1, …, N. A consistent interpolation operator L , as defined in this paper, interpolates these function values for N new abscissas t_n = (n+½)Δt+t₀, the first N?1 of which are halfway between those originally given while the last one is outside of the original abscissa range. Application of L to these interpolated function values produces the last N?1 samples u₂, u₃, …, u_N of the original data plus one extrapolated function value u_N+1. Hence, L ² is essentially a shift operator, but with a prediction component. The difference between various interpolation methods (e.g. polynomials, Fourier series) is now reduced to the way in which u_N+1 is determined. This concept not only permits a uniform view at interpolation by quite different classes of functions but also allows the creation of more general interpolation, differentiation, and integration formulas, which can be tailored to particular problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as _∂ttu−Δu=f(x,u) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as t_∂u−Δu=f(x,u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.     

Global well‐posedness for strongly damped multidimensional generalized Boussinesq equations

We study the Cauchy problem for a class of strongly damped multidimensional generalized Boussinesq equations u_tt?Δu?Δu_tt+Δ²u+Δ²u_tt?kΔu_t=Δf(u), where k is a positive constant. Under some assumptions and by using potential well method, we prove the existence and nonexistence of global weak solution without solution without establishing the local existence theory. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown parameters

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u_t(x, t)  = u_xx + qu_x(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ₀, u(1, t) = ψ₁. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H^1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data u_x(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H^1,2[0,T] is given explicitly interms of the semigroup.     

Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction-diffusion system

We establish the existence of standing waves with one pulse, multiple spikes and transition layers in the nonlinear reaction-diffusion system

u_t=f(u,w)+u_xx,w_t=?²g(u,w)+w_xx,x∈R,     

The Instability of Nonmonotonic Wave Solutions of Parabolic Equations

We consider the class of equations u_t=f(u_xx, u_x, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of u_t = F(u_xx, u_xy, u_x, u_y, u). Finally, we consider the class of equations u_t=f(u_xx, u_x, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x).     

Positive Steady-state Solutions of a Non-linear Reaction–Diffusion System

The steady-state problem of the non-linear reaction-diffusion system u_t−u_xx=u(av−b) , v_t−v_xx=cu is considered. The existence of positive steady-solutions is established by using a fixed point theorem in ordered Banach space. The uniqueness of ordered positive steady-state solutions and an application to the associated reaction-diffusion system are presented. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Small generalized breathers with exponentially small tails for Klein–Gordon equations

We consider a class of nonlinear Klein–Gordon equation _utt=_uxx−u+f(u)$u_{tt}=u_{xx}-u+f(u)$ and show that generically there exist small breathers with exponentially small tails.     

On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition

We establish the critical Fujita exponents for the solution of the porous medium equation u_t=Δu^m, x∈R₊^N, t>0, subject to the nonlinear boundary condition −∂u^m/∂x₁=u^p, x₁=0, t>0, in multi-dimension.     

A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms

u_tt−Δu−γ(t)Δu_t+β_ε(t)u_t=f(u),     

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.     

Boundedness of global positive solutions of a porous medium equation with a moving localized source

In this paper a porous medium equation with a moving localized source ut=ur(Δu+af(u(x₀(t),t))) is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, in one space dimension case, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.     

Regularizing properties and propagations of singularities for thermoelastic plates

We consider the thermoelastic plate system, u_tt−γΔu_tt+Δ²u+αΔθ=0 , θ_t−κΔθ−αΔu_t=0 and we make a comparison between the models in which γ=0 and γ>0. We conclude that in the first case the plate system is of a parabolic type, while when γ>0 the corresponding system has a hyperbolic behaviour. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Alcune limitazioni per le soluzioni di equazioni paraboliche

Sunto Si discute il problema diCauchy relativo all'equazione del calore, u_t=u_xx, ed a condizioni iniziali sull'asse t. Si provano inoltre alcune proprietà delle soluzioni positive di tale problema e di analoghi problemi diCauchy relativi a più generali equazioni paraboliche.

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Summary We consider theCauchy problem for the heat equation, u_t=u_xx, with initial conditions on the t-axis. We prove some property of positive solutions of this problem, and of similarCauchy problems for more general parabolic equations.

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A Giovanni Sansone nel suo 70^mo compleanno.