—Aim of this paper is the qualitative analysis of a boundary value problem for a third order non ... more —Aim of this paper is the qualitative analysis of a boundary value problem for a third order non linear parabolic equation which describes several dissipative models. When the source term is linear, the problem is explictly solved by means of a Fourier series with properties of rapid convergence. In the non linear case, appropriate estimates of this series allow to deduce the asymp-totic behaviour of the solution.
A boundary value problem P ε related to a third-order parabolic equation with a small parameter ε... more A boundary value problem P ε related to a third-order parabolic equation with a small parameter ε is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third-order parabolic operator regularizes various non linear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution of P ε is estimated by means of slow time τ = εt and fast time θ = t/ε. As consequence, a rigorous asymptotic approximation for the solution of P ε is established. partial different equations / viscoelasticity/ superconductivity/ boundary layer Diffusion et comportement ondullux dans le modél linear de Voigt Résumé. On analyse unprobì eme P ε des valeurs au contour relativementà une equation parabolique dutroisì eme ordre. Cette equation regle l'evolution unidimensionnel de beaucoup de materiels dissipatifs comme le fluides ou les solides visquelastiques, les gaz visqueux, les materiels superconductibles, les fluides incompressiblesélectriquement conductibles. De plus l'opérateur parabolique dutroisì eme ordre regularise divers equations non lineaires des ondes dudeuxì eme ordre. On examine dans ce travail le comportment hy-perbolique ou parabolique de la solution du P ε moyennant le temps lent et le temps rapide. En conséquence, on pose une rigoureuse approximation asymptotique pour la solution du P ε. ´ equations aux dérivées partielles / viscollasticité / supraconduc-tivité
The linear Kelvin-Voigt operator Lε is a typical example of wave operator L 0 perturbed by higher... more The linear Kelvin-Voigt operator Lε is a typical example of wave operator L 0 perturbed by higher-order viscous terms as εuxxt. If Pε is a prefixed boundary-value problem for Lε, when ε = 0 Lε turns into L 0 and Pε into a problem P 0 with the same initial-boundary conditions of Pε. Boundary-layers are missing and the related control terms depending on the fast time are neglegible. In a small time-interval, the wave behavior is a realistic approximation of uε when ε → 0. On the contrary, when t is large, diffusion effects should prevail and the behavior of uε for ε → 0 and t → ∞ should be analyzed. For this, a suitable functional corrispondence between the Green functions Gε and G 0 of Pε and P 0 is achieved and its asymptotic behavior is rigorously examined. For this, a suitable functional corrispondence between the Green functions Gε and G 0 of Pε and P 0 is derived and its asymptotic behavior is rigorously examined. As consequence, the interaction between diffusion effects and pure waves is evaluated by means of the slow time ε t; the main results show that in time-intervals as (ε, 1/ε) pure waves are propagated nearly undisturbed, while damped oscillations predominate as from the instant t > 1/ε.
An evolution operator Ln with n arbitrary, typical of several models, is analyzed. When n = 1 the... more An evolution operator Ln with n arbitrary, typical of several models, is analyzed. When n = 1 the operator characterizes the Standard Linear Solid of viscoelasticity, whose properties are already extablished in previous papers. The fundamental solution En of Ln is explictly obtained and it's estimated in terms of the fundamental solution E1 of L1. So, whatever n may be, asymptotic properties and maximum theorems are achieved. These results are applied to the Rouse model and reptation model, which describe different aspects of polymer chains.
The paper deals with the explicit calculus and the properties of the fundamental solution K of a ... more The paper deals with the explicit calculus and the properties of the fundamental solution K of a parabolic operator related to a semilinear equation that models reaction diffusion systems with excitable kinetics. The initial value problem in all of the space is analyzed together with continuous dependence and a priori estimates of the solution. These estimates show that the asymptotic behavior is determined by the reaction mechanism. Moreover it's possible a rigorous singular perturbation analysis for discussing travelling waves with their characteristic times.
The paper deals with a third order semilinear equation which characterizes exponentially shaped J... more The paper deals with a third order semilinear equation which characterizes exponentially shaped Josephson junctions in superconductivity. The initial-boundary problem with Dirichlet conditions is analyzed. When the source term F is a linear function, the problem is explicitly solved by means of a Fourier series with properties of rapid convergence. When F is nonlin-ear, appropriate estimates of this series allow to deduce a priori estimates, continuous dependence and asymptotic behaviour of the solution.
The reaction-diffusion system of Fitzhugh Nagumo is considered. The initial-boundary problems wit... more The reaction-diffusion system of Fitzhugh Nagumo is considered. The initial-boundary problems with Neumann and Dirichlet conditions are analyzed. By means of an equivalent semilinear integrodifferential equation which characterizes several dissipative models of viscoelasticity, biology, and superconductivity, some results on existence, uniqueness and a priori estimates are deduced both in the linear case and in the non linear one.
We prove existence and uniqueness of solutions of a large class of initial-boundary-value problem... more We prove existence and uniqueness of solutions of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation (the third order term being dissipative) on a finite space interval with Dirichlet, Neumann or pseudope-riodic boundary conditions. The class includes equations arising in superconductor theory, such as a well-known modified sine-Gordon equation describing the Josephson effect, and in the theory of viscoelastic materials.
The paper deals with an integrodifferential operator which models numerous phenomena in supercond... more The paper deals with an integrodifferential operator which models numerous phenomena in superconductivity, in biology and in viscoelasticity. Initial-boundary value problems with Neumann, Dirichlet and mixed boundary conditions are analyzed. An asymptotic analysis is achieved proving that for large t, the influences of the initial data vanish, while the effects of boundary disturbances are everywhere bounded.
A Neumann problem in the strip for the Fitzhugh Nagumo system is considered. The transformation i... more A Neumann problem in the strip for the Fitzhugh Nagumo system is considered. The transformation in a non linear integral equation permits to deduce a priori estimates for the solution. A complete asymptotic analysis shows that for large t the effects of the initial data vanish while the effects of boundary disturbances ϕ 1 (t), ϕ 2 (t) depend on the properties of the data. When ϕ 1 , ϕ 2 are convergent for large t , the solution is everywhere bounded; when ˙ ϕ i ∈ L 1 (0, ∞)(i = 1, 2) too, the effects are vanishing.
A superconductive model characterized by a third order parabolic operator L ε is analysed. When t... more A superconductive model characterized by a third order parabolic operator L ε is analysed. When the viscous terms, represented by higher-order derivatives , tend to zero, a hyperbolic operator L 0 appears. Furthermore, if P ε is the Dirichlet initial boundary-value problem for L ε , when L ε turns into L 0 , P ε turns into a problem P 0 with the same initial-boundary conditions as P ε. The solution of the nonlinear problem related to the remainder term r is achieved, as long as the higher-order derivatives of the solution of P 0 are bounded. Moreover , some classes of explicit solutions related to P 0 are determined, proving the existence of at least one motion whose derivatives are bounded. The estimate shows that the diffusion effects are bounded even when time tends to infinity. PACS 74.45.+c AMS 35K35,35E35
A parabolic integro differential operator L, suitable to describe many phenomena in various physi... more A parabolic integro differential operator L, suitable to describe many phenomena in various physical fields, is considered. By means of equivalence between L and the third order equation describing the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, explicitly evaluating, boundary contributions related to the Dirichlet problem.
A parabolic integro differential operator operator L suitable to describe many phenomena in vario... more A parabolic integro differential operator operator L suitable to describe many phenomena in various physical fields,is considered. By means of equivalence between L and the third order equation which describe the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, evaluating explicitly boundary contributions related to the Dirichlet problem.
An integro differential equation which is able to describe the evolution of a large class of diss... more An integro differential equation which is able to describe the evolution of a large class of dissipative models, is considered. By means of an equivalence, the focus shifts to the perturbed sine-Gordon equation that in superconductiv-ity finds interesting applications in multiple engineering areas. The Neumann boundary problem is considered, and the behaviour of a viscous term, defined by a higher-order derivative with small diffusion coefficient ε, is investigated. The Green function, expressed by means of Fourier series, is considered, and an estimate is achieved. Furthermore, some classes of solutions of the hyperbolic equation are determined, proving that there exists at least one solution with bounded derivatives. Results obtained prove that diffusion effects are bounded and tend to zero when ε tends to zero.
Mathematical models related to some Josephson junctions are pointed out and attention is drawn to... more Mathematical models related to some Josephson junctions are pointed out and attention is drawn to the solutions of certain initial boundary problems and to some of their estimates. In addition, results of rigorous analysis of the behaviour of these solutions when t → ∞ and when the small parameter ε tends to zero are cited. These analyses lead us to mention some of the open problems.
A Neumann problem for a wave equation perturbed by viscous terms with small parameters is conside... more A Neumann problem for a wave equation perturbed by viscous terms with small parameters is considered. The interaction of waves with the diffusion effects caused by a higher-order derivative with small coefficient ε, is investigated. Results obtained prove that for slow time εt < 1 waves are propagated almost undisturbed, while for fast time t > 1 ε diffusion effects prevail.
—Aim of this paper is the qualitative analysis of a boundary value problem for a third order non ... more —Aim of this paper is the qualitative analysis of a boundary value problem for a third order non linear parabolic equation which describes several dissipative models. When the source term is linear, the problem is explictly solved by means of a Fourier series with properties of rapid convergence. In the non linear case, appropriate estimates of this series allow to deduce the asymp-totic behaviour of the solution.
A boundary value problem P ε related to a third-order parabolic equation with a small parameter ε... more A boundary value problem P ε related to a third-order parabolic equation with a small parameter ε is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third-order parabolic operator regularizes various non linear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution of P ε is estimated by means of slow time τ = εt and fast time θ = t/ε. As consequence, a rigorous asymptotic approximation for the solution of P ε is established. partial different equations / viscoelasticity/ superconductivity/ boundary layer Diffusion et comportement ondullux dans le modél linear de Voigt Résumé. On analyse unprobì eme P ε des valeurs au contour relativementà une equation parabolique dutroisì eme ordre. Cette equation regle l'evolution unidimensionnel de beaucoup de materiels dissipatifs comme le fluides ou les solides visquelastiques, les gaz visqueux, les materiels superconductibles, les fluides incompressiblesélectriquement conductibles. De plus l'opérateur parabolique dutroisì eme ordre regularise divers equations non lineaires des ondes dudeuxì eme ordre. On examine dans ce travail le comportment hy-perbolique ou parabolique de la solution du P ε moyennant le temps lent et le temps rapide. En conséquence, on pose une rigoureuse approximation asymptotique pour la solution du P ε. ´ equations aux dérivées partielles / viscollasticité / supraconduc-tivité
The linear Kelvin-Voigt operator Lε is a typical example of wave operator L 0 perturbed by higher... more The linear Kelvin-Voigt operator Lε is a typical example of wave operator L 0 perturbed by higher-order viscous terms as εuxxt. If Pε is a prefixed boundary-value problem for Lε, when ε = 0 Lε turns into L 0 and Pε into a problem P 0 with the same initial-boundary conditions of Pε. Boundary-layers are missing and the related control terms depending on the fast time are neglegible. In a small time-interval, the wave behavior is a realistic approximation of uε when ε → 0. On the contrary, when t is large, diffusion effects should prevail and the behavior of uε for ε → 0 and t → ∞ should be analyzed. For this, a suitable functional corrispondence between the Green functions Gε and G 0 of Pε and P 0 is achieved and its asymptotic behavior is rigorously examined. For this, a suitable functional corrispondence between the Green functions Gε and G 0 of Pε and P 0 is derived and its asymptotic behavior is rigorously examined. As consequence, the interaction between diffusion effects and pure waves is evaluated by means of the slow time ε t; the main results show that in time-intervals as (ε, 1/ε) pure waves are propagated nearly undisturbed, while damped oscillations predominate as from the instant t > 1/ε.
An evolution operator Ln with n arbitrary, typical of several models, is analyzed. When n = 1 the... more An evolution operator Ln with n arbitrary, typical of several models, is analyzed. When n = 1 the operator characterizes the Standard Linear Solid of viscoelasticity, whose properties are already extablished in previous papers. The fundamental solution En of Ln is explictly obtained and it's estimated in terms of the fundamental solution E1 of L1. So, whatever n may be, asymptotic properties and maximum theorems are achieved. These results are applied to the Rouse model and reptation model, which describe different aspects of polymer chains.
The paper deals with the explicit calculus and the properties of the fundamental solution K of a ... more The paper deals with the explicit calculus and the properties of the fundamental solution K of a parabolic operator related to a semilinear equation that models reaction diffusion systems with excitable kinetics. The initial value problem in all of the space is analyzed together with continuous dependence and a priori estimates of the solution. These estimates show that the asymptotic behavior is determined by the reaction mechanism. Moreover it's possible a rigorous singular perturbation analysis for discussing travelling waves with their characteristic times.
The paper deals with a third order semilinear equation which characterizes exponentially shaped J... more The paper deals with a third order semilinear equation which characterizes exponentially shaped Josephson junctions in superconductivity. The initial-boundary problem with Dirichlet conditions is analyzed. When the source term F is a linear function, the problem is explicitly solved by means of a Fourier series with properties of rapid convergence. When F is nonlin-ear, appropriate estimates of this series allow to deduce a priori estimates, continuous dependence and asymptotic behaviour of the solution.
The reaction-diffusion system of Fitzhugh Nagumo is considered. The initial-boundary problems wit... more The reaction-diffusion system of Fitzhugh Nagumo is considered. The initial-boundary problems with Neumann and Dirichlet conditions are analyzed. By means of an equivalent semilinear integrodifferential equation which characterizes several dissipative models of viscoelasticity, biology, and superconductivity, some results on existence, uniqueness and a priori estimates are deduced both in the linear case and in the non linear one.
We prove existence and uniqueness of solutions of a large class of initial-boundary-value problem... more We prove existence and uniqueness of solutions of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation (the third order term being dissipative) on a finite space interval with Dirichlet, Neumann or pseudope-riodic boundary conditions. The class includes equations arising in superconductor theory, such as a well-known modified sine-Gordon equation describing the Josephson effect, and in the theory of viscoelastic materials.
The paper deals with an integrodifferential operator which models numerous phenomena in supercond... more The paper deals with an integrodifferential operator which models numerous phenomena in superconductivity, in biology and in viscoelasticity. Initial-boundary value problems with Neumann, Dirichlet and mixed boundary conditions are analyzed. An asymptotic analysis is achieved proving that for large t, the influences of the initial data vanish, while the effects of boundary disturbances are everywhere bounded.
A Neumann problem in the strip for the Fitzhugh Nagumo system is considered. The transformation i... more A Neumann problem in the strip for the Fitzhugh Nagumo system is considered. The transformation in a non linear integral equation permits to deduce a priori estimates for the solution. A complete asymptotic analysis shows that for large t the effects of the initial data vanish while the effects of boundary disturbances ϕ 1 (t), ϕ 2 (t) depend on the properties of the data. When ϕ 1 , ϕ 2 are convergent for large t , the solution is everywhere bounded; when ˙ ϕ i ∈ L 1 (0, ∞)(i = 1, 2) too, the effects are vanishing.
A superconductive model characterized by a third order parabolic operator L ε is analysed. When t... more A superconductive model characterized by a third order parabolic operator L ε is analysed. When the viscous terms, represented by higher-order derivatives , tend to zero, a hyperbolic operator L 0 appears. Furthermore, if P ε is the Dirichlet initial boundary-value problem for L ε , when L ε turns into L 0 , P ε turns into a problem P 0 with the same initial-boundary conditions as P ε. The solution of the nonlinear problem related to the remainder term r is achieved, as long as the higher-order derivatives of the solution of P 0 are bounded. Moreover , some classes of explicit solutions related to P 0 are determined, proving the existence of at least one motion whose derivatives are bounded. The estimate shows that the diffusion effects are bounded even when time tends to infinity. PACS 74.45.+c AMS 35K35,35E35
A parabolic integro differential operator L, suitable to describe many phenomena in various physi... more A parabolic integro differential operator L, suitable to describe many phenomena in various physical fields, is considered. By means of equivalence between L and the third order equation describing the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, explicitly evaluating, boundary contributions related to the Dirichlet problem.
A parabolic integro differential operator operator L suitable to describe many phenomena in vario... more A parabolic integro differential operator operator L suitable to describe many phenomena in various physical fields,is considered. By means of equivalence between L and the third order equation which describe the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, evaluating explicitly boundary contributions related to the Dirichlet problem.
An integro differential equation which is able to describe the evolution of a large class of diss... more An integro differential equation which is able to describe the evolution of a large class of dissipative models, is considered. By means of an equivalence, the focus shifts to the perturbed sine-Gordon equation that in superconductiv-ity finds interesting applications in multiple engineering areas. The Neumann boundary problem is considered, and the behaviour of a viscous term, defined by a higher-order derivative with small diffusion coefficient ε, is investigated. The Green function, expressed by means of Fourier series, is considered, and an estimate is achieved. Furthermore, some classes of solutions of the hyperbolic equation are determined, proving that there exists at least one solution with bounded derivatives. Results obtained prove that diffusion effects are bounded and tend to zero when ε tends to zero.
Mathematical models related to some Josephson junctions are pointed out and attention is drawn to... more Mathematical models related to some Josephson junctions are pointed out and attention is drawn to the solutions of certain initial boundary problems and to some of their estimates. In addition, results of rigorous analysis of the behaviour of these solutions when t → ∞ and when the small parameter ε tends to zero are cited. These analyses lead us to mention some of the open problems.
A Neumann problem for a wave equation perturbed by viscous terms with small parameters is conside... more A Neumann problem for a wave equation perturbed by viscous terms with small parameters is considered. The interaction of waves with the diffusion effects caused by a higher-order derivative with small coefficient ε, is investigated. Results obtained prove that for slow time εt < 1 waves are propagated almost undisturbed, while for fast time t > 1 ε diffusion effects prevail.
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