2,303 research outputs found
Reaction-diffusion problems on time-dependent Riemannian manifolds: stability of periodic solutions
We investigate the stability of time-periodic solutions of semilinear
parabolic problems with Neumann boundary conditions. Such problems are posed on
compact submanifolds evolving periodically in time. The discussion is based on
the principal eigenvalue of periodic parabolic operators. The study is
motivated by biological models on the effect of growth and curvature on
patterns formation. The Ricci curvature plays an important role
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries
In this work we study the behavior of a family of solutions of a semilinear
elliptic equation, with homogeneous Neumann boundary condition, posed in a
two-dimensional oscillating thin region with reaction terms concentrated in a
neighborhood of the oscillatory boundary. Our main result is concerned with the
upper and lower semicontinuity of the set of solutions. We show that the
solutions of our perturbed equation can be approximated with ones of a
one-dimensional equation, which also captures the effects of all relevant
physical processes that take place in the original problem
Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDG method
In our earlier work [8], we approximated solutions of a general class of
scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous
Galerkin (Interpolatory HDG) method. This method reduces the computational cost
compared to standard HDG since the HDG matrices are assembled once before the
time integration. Interpolatory HDG also achieves optimal convergence rates;
however, we did not observe superconvergence after an element-by-element
postprocessing. In this work, we revisit the Interpolatory HDG method for
reaction diffusion problems, and use the postprocessed approximate solution to
evaluate the nonlinear term. We prove this simple change restores the
superconvergence and keeps the computational advantages of the Interpolatory
HDG method. We present numerical results to illustrate the convergence theory
and the performance of the method
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
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