Uniqueness of bounded solutions for some abstract differential equations

Summary For some differential equations in Hilbert space we prove uniqueness of bounded solutions in the Stepanoff norm.

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Riassunto Si dimostra l’unicità delle soluzioniS ²-limitate per alcune equazioni differenziali astratte.

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This research is supported by a grant of the N.R.C. of Canada.     

Uniqueness of bounded weak solutions for some abstract differential equations

Riassunto Si dà un teorema di unicità per soluzioni deboli limitate di alcune equazioni differenziali astratte.

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This research is supported by of the N.R.C. of Canada.     

Uniqueness and continuous dependence results for solutions of singular differential equations

Solutions of Cauchy problems for the singular equations $\text{u}{}_{\mathrm{tt}}\text{+ (}\text{Ψ(t)}\text{t}\text{) u}{}_{\mathrm{t}}\text{= Mu}$ (in a Hilbert space setting) and $\text{u}{}_{\mathrm{t}}\text{+ Δu +}\text{∑}\text{m}\text{i=1}\text{((}\text{k}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)(}\text{?}{}_{\mathrm{i}}\text{?}{}_{\mathrm{i}}\text{)) + g(t)u=0}\text{in}\text{ω × |0,T)}$, ω={(x₁,…,x_M)ε$\text{R}$^m: 0 < x_i < c_i for each i=1,…,m} are shown to be unique and to depend Hölder continuously on the initial data in suitably chosen measures for 0?t < T < ∞. Logarithmic convexity arguments are used to derive the inequalities from which such results can be deduced.     

On global regular solutions of third order partial differential equations

Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. Diffusion in a media with two distinct families of diffusion paths is modelled by two coupled linear partial differential equations of parabolic type with diffusivities D₁ and D₂. Physically the situation D₂ ? D₁ is of some considerable interest and previously established results, for D₂ non-zero, for the solution of boundary value problems, are not applicable to the idealized theory characterized by D₂ vanishing. An integral equation, which arises in the solution of boundary value problems for this idealized theory, is formally solved.     

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Uniqueness for some Leray-Hopf solutions to the Navier-Stokes equations

We exhibit simple sufficient conditions which give weak-strong uniqueness for the 3D Navier-Stokes equations. The main tools are trilinear estimates and energy inequalities. We then apply our result to the framework of Lorentz, Morrey and Besov over Morrey spaces so as to get new weak-strong uniqueness classes and so uniqueness classes for solutions in the Leray-Hopf class. In the last section, we give a uniqueness and regularity result. We obtain new uniqueness classes for solutions in the Leray-Hopf class without energy inequalities but sufficiently regular.     

Uniqueness and growth of weak solutions to certain linear differential equations in Hilbert space

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Series solutions of coupled differential equations with one regular singular point

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In this paper, we present an extended Method of Frobenius on a coupled system of two ordinary differential equations. These equations come from the micropolar theory, which is one of the three kinds of the new 3M physics.     

Uniqueness of positive solutions of a class of quasilinear ordinary differential equations

Abstract. Uniqueness results are obtained for positive solutions of a class of quasilinear ordi-nary differential equations. The methods rely on the energy analysis and a scale argument.     

Global differential equations for slice regular functions

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