scholarly journals Random Evolution Equations: Well-Posedness, Asymptotics, and Applications to Graphs

2021 ◽  
Author(s):  
Stefano Bonaccorsi ◽  
Francesca Cottini ◽  
Delio Mugnolo
Keyword(s):  
Evolution Equations ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Difference Operators ◽  
Well Posedness ◽  
Metric Graphs ◽  
Diffusion Type ◽  
Convergence In Norm ◽  
Combinatorial Graphs

AbstractWe study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findings in two environments with randomly evolving features: ensembles of difference operators on combinatorial graphs, or else of differential operators on metric graphs.

On invariant subspaces for nonlinear finite-difference operators

1998 ◽  
Vol 128 (6) ◽  
pp. 1293-1308 ◽  
Author(s):  
Victor A. Galaktionov
Keyword(s):  
Finite Difference ◽  
Differential Operators ◽  
Invariant Subspaces ◽  
Reaction Diffusion ◽  
Difference Operators ◽  
Diffusion Type ◽  
Nonlinear Differential Operators ◽  
Discrete Operators ◽  
Spatial Discretisation ◽  
Lower Dimensional

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

scholarly journals Linear Hyperbolic Systems on Networks: well-posedness and qualitative properties

2020 ◽  
Author(s):  
Marjeta Kramar ◽  
Delio Mugnolo ◽  
Serge Nicaise
Keyword(s):  
Boundary Conditions ◽  
Evolution Equations ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Order Reduction ◽  
Well Posedness ◽  
Qualitative Properties ◽  
Local Boundary ◽  
First Order ◽  
Necessary And Sufficient

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first - order reduction . We study forward and backward well - posedness ; furthermore , we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first - order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied . p, li { white-space: pre-wrap; }

scholarly journals Eventual domination for linear evolution equations

2021 ◽  
Author(s):  
Jochen Glück ◽  
Delio Mugnolo
Keyword(s):  
Evolution Equations ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Abstract Theory ◽  
Important Special Case ◽  
Linear Evolution ◽  
Necessary And Sufficient ◽  
Adjoint Operators ◽  
Linear Evolution Equations ◽  
Special Case

AbstractWe consider two $$C_0$$ C 0 -semigroups on function spaces or, more generally, Banach lattices and give necessary and sufficient conditions for the orbits of the first semigroup to dominate the orbits of the second semigroup for large times. As an important special case we consider an $$L^2$$ L 2 -space and self-adjoint operators A and B which generate $$C_0$$ C 0 -semigroups; in this situation we give criteria for the existence of a time $$t_1 \ge 0$$ t 1 ≥ 0 such that $$e^{tB} \ge e^{tA}$$ e tB ≥ e tA for all subsequent times $$t\ge t_1$$ t ≥ t 1 . As a consequence of our abstract theory, we obtain many surprising insights into the behaviour of various second and fourth order differential operators.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

scholarly journals Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Denghao Pang ◽  
Wei Jiang ◽  
Azmat Ullah Khan Niazi ◽  
Jiale Sheng
Keyword(s):  
Banach Spaces ◽  
Mild Solution ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Well Posedness ◽  
Optimal Controls ◽  
Lagrange Problem ◽  
Fractional Differential ◽  
Fractional Evolution Systems ◽  
Definition Of

AbstractIn this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.

The weighted Cauchy problem for linear functional differential equations with strong singularities

2011 ◽  
Vol 18 (3) ◽  
pp. 577-586
Author(s):  
Zaza Sokhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Initial Point ◽  
Higher Order ◽  
Well Posedness ◽  
Deviating Arguments ◽  
Functional Differential

Abstract The sufficient conditions of well-posedness of the weighted Cauchy problem for higher order linear functional differential equations with deviating arguments, whose coefficients have nonintegrable singularities at the initial point, are found.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

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