620 research outputs found
Hyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions
Under consideration is the hyperbolic relaxation of a semilinear
reaction-diffusion equation on a bounded domain, subject to a dynamic boundary
condition. We also consider the limit parabolic problem with the same dynamic
boundary condition. Each problem is well-posed in a suitable phase space where
the global weak solutions generate a Lipschitz continuous semiflow which admits
a bounded absorbing set. We prove the existence of a family of global
attractors of optimal regularity. After fitting both problems into a common
framework, a proof of the upper-semicontinuity of the family of global
attractors is given as the relaxation parameter goes to zero. Finally, we also
establish the existence of exponential attractors.Comment: to appear in Quarterly of Applied Mathematic
On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds
We investigate existence and nonexistence of stationary stable nonconstant
solutions, i.e. patterns, of semilinear parabolic problems in bounded domains
of Riemannian manifolds satisfying Robin boundary conditions. These problems
arise in several models in applications, in particular in Mathematical Biology.
We point out the role both of the nonlinearity and of geometric objects such as
the Ricci curvature of the manifold, the second fundamental form of the
boundary of the domain and its mean curvature. Special attention is devoted to
surfaces of revolution and to spherically symmetric manifolds, where we prove
refined results
Slowly Oscillating Solution of the Cubic Heat Equation
In this paper, we are considering the Cauchy problem of the nonlinear heat
equation . After extending Y. Meyer's
result establishing the existence of global solutions, under a smallness
condition of the initial data in the homogeneous Besov spaces
, where 3 \textless{} p
\textless{} 9 and , we prove that initial data , arbitrarily small in , can produce solutions that explode in
finite time. In addition, the blowup may occur after an arbitrarily short time
On a Class of Energy Preserving Boundary Conditions for Incompressible Newtonian Flows
We derive a class of energy preserving boundary conditions for incompressible
Newtonian flows and prove local-in-time well-posedness of the resulting initial
boundary value problems, i.e. the Navier-Stokes equations complemented by one
of the derived boundary conditions, in an Lp-setting in domains, which are
either bounded or unbounded with almost flat, sufficiently smooth boundary. The
results are based on maximal regularity properties of the underlying
linearisations, which are also established in the above setting.Comment: 53 page
Semi-group theory for the Stokes operator with Navier-type boundary conditions on -spaces
In this article we consider the Stokes problem with Navier-type boundary
conditions on a domain , not necessarily simply connected. Since under
these conditions the Stokes problem has a non trivial kernel, we also study the
solutions lying in the orthogonal of that kernel. We prove the analyticity of
several semigroups generated by the Stokes operator considered in different
functional spaces. We obtain strong, weak and very weak solutions for the time
dependent Stokes problem with the Navier-type boundary condition under
different hypothesis on the initial data and external force
. Then, we study the fractional and pure imaginary powers of
several operators related with our Stokes operators. Using the fractional
powers, we prove maximal regularity results for the homogeneous Stokes problem.
On the other hand, using the boundedness of the pure imaginary powers we deduce
maximal regularity for the inhomogeneous Stokes problem
On the third boundary value problem for parabolic equations in a non-regular domain of Rá´º +1
In this paper, we look for sufficient conditions on the lateral surface of the domain and on the coefficients of the boundary conditions of a N−space dimensional
linear parabolic equation, in order to obtain existence, uniqueness and maximal regularity of the solution in a Hilbertian anisotropic Sobolev space when the right hand side of the equation is in a Lebesgue space. This work is an extension of solvability results obtained for a second order parabolic equation, set in a non-regular domain of R 3 obtained in [1],
to the case where the domain is cylindrical, not with respect to the time variable, but with respect to N space variables, N > 1.Publisher's Versio
Optimal Control of the Thermistor Problem in Three Spatial Dimensions
This paper is concerned with the state-constrained optimal control of the
three-dimensional thermistor problem, a fully quasilinear coupled system of a
parabolic and elliptic PDE with mixed boundary conditions. This system models
the heating of a conducting material by means of direct current. Local
existence, uniqueness and continuity for the state system are derived by
employing maximal parabolic regularity in the fundamental theorem of Pr\"uss.
Global solutions are addressed, which includes analysis of the linearized state
system via maximal parabolic regularity, and existence of optimal controls is
shown if the temperature gradient is under control. The adjoint system
involving measures is investigated using a duality argument. These results
allow to derive first-order necessary conditions for the optimal control
problem in form of a qualified optimality system. The theoretical findings are
illustrated by numerical results
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