27 research outputs found
Blow-up for the Porous Media Equation with Source Term and Positive Initial Energy
AbstractWe study the Cauchy–Dirichlet problem for the porous media equation with nonlinear source term in a bounded subset of Rn. The problem describes the propagation of thermal perturbations in a medium with a nonlinear heat-conduction coefficient and a heat source depending on the temperature. The aim of the paper is to extend the unstable set to a part of the positive energy region, a phenomenon which was known only for linear conduction
Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition
The paper deals with the existence and multiplicity of nontrivial solutions
for the doubly elliptic problem where is a
bounded open subset of () with boundary
, ,
being nonempty and relatively open on ,
and being subcritical with respect to
Sobolev embedding on .
We prove that the problem admits nontrivial solutions at the potential--well
depth energy level, which is the minimal energy level for nontrivial solutions.
We also prove that the problem has infinitely many solutions at higher energy
levels
The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces
The aim of the paper is to study the problem
where
is a open domain of with uniformly boundary
(, ), , is a
relatively open partition of with (but not )
possibly empty. Here and denote the
Riemannian divergence and gradient operators on , is the outward
normal to , the coefficients are
suitably regular functions on with and uniformly
positive, is a suitably regular function in and is a positive
constant.
In this paper we first study well-posedness in the natural energy space and
give regularity results. Hence we study asymptotic stability for solutions when
is bounded, is connected, , is constant and
.Comment: The paper also extends some results in arXiv:2105.0921
Klein-Gordon-Maxwell equations driven by mixed local-nonlocal operators
Classical results concerning Klein-Gordon-Maxwell type systems are shortly
reviewed and generalized to the setting of mixed local-nonlocal operators,
where the nonlocal one is allowed to be nonpositive definite according to a
real parameter. In this paper, we provide a range of parameter values to ensure
the existence of solitary (standing) waves, obtained as Mountain Pass critical
points for the associated energy functionals in two different settings, by
considering two different classes of potentials: constant potentials and
continuous, bounded from below, and coercive potentials.Comment: 26 pages, 2 figure
Schr\"odinger-Maxwell equations driven by mixed local-nonlocal operators
In this paper we prove existence of solutions to Schr\"odinger-Maxwell type
systems involving mixed local-nonlocal operators. Two different models are
considered: classical Schr\"odinger-Maxwell equations and Schr\"odinger-Maxwell
equations with a coercive potential, and the main novelty is that the nonlocal
part of the operator is allowed to be nonpositive definite according to a real
parameter. We then provide a range of parameter values to ensure the existence
of solitary standing waves, obtained as Mountain Pass critical points for the
associated energy functionals.Comment: arXiv admin note: text overlap with arXiv:2303.1166
Heat equation with dynamical boundary conditions of reactive-diffusive type
This paper deals with the heat equation posed in a bounded regular domain
coupled with a dynamical boundary condition of reactive-diffusive type,
involving the Laplace-Beltrami operator. We prove well-posedness of the problem
together with regularity of the solutions.Comment: 18 page