111 research outputs found
Supersolutions for a class of semilinear heat equations
A semilinear heat equation with nonnegative initial
data in a subset of is considered under the assumption that
is nonnegative and nondecreasing and . A simple
technique for proving existence and regularity based on the existence of
supersolutions is presented, then a method of construction of local and global
supersolutions is proposed. This approach is applied to the model case
, : new sufficient conditions for the
existence of local and global classical solutions are derived in the critical
and subcritical range of parameters. Some possible generalisations of the
method to a broader class of equations are discussed.Comment: Expanded version of the previous submission arXiv:1111.0258v1. 14
page
No Touchdown at Zero Points of the Permittivity Profile for the MEMS Problem
[[abstract]]We study the quenching behavior for a semilinear heat equation arising in models of micro-electro-mechanical systems (MEMS). The problem involves a source term with a spatially dependent potential, given by the dielectric permittivity profile, and quenching corresponds to a touchdown phenomenon. It is well known that quenching does occur. We prove that touchdown cannot occur at zero points of the permittivity profile. In particular, we remove the assumption of compactness of the touchdown set, made in all previous work on the subject and whose validity is unknown in most typical cases. This answers affirmatively a conjecture made in [W. Guo, Z. Pan, and M. J. Ward, SIAM J. Appl. Math., 66 (2005), pp. 309--338] on the basis of numerical evidence. The result crucially depends on a new type I estimate of the quenching rate, that we establish. In addition we obtain some sufficient conditions for compactness of the touchdown set, without a convexity assumption on the domain. These results may be of some qualitative importance in applications to MEMS optimal design, especially for devices such as microvalves.[[notice]]補正完畢[[journaltype]]國外[[incitationindex]]SCI[[ispeerreviewed]]Y[[booktype]]紙本[[countrycodes]]US
A new critical curve for the Lane-Emden system
We study stable positive radially symmetric solutions for the Lane-Emden
system in , in , where .
We obtain a new critical curve that optimally describes the existence of such
solutions.Comment: 13 pages, 1 figur
Non-existence and uniqueness results for supercritical semilinear elliptic equations
Non-existence and uniqueness results are proved for several local and
non-local supercritical bifurcation problems involving a semilinear elliptic
equation depending on a parameter. The domain is star-shaped but no other
symmetry assumption is required. Uniqueness holds when the bifurcation
parameter is in a certain range. Our approach can be seen, in some cases, as an
extension of non-existence results for non-trivial solutions. It is based on
Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise
in many applications, for instance in astrophysics, hydrodynamics or
thermodynamics. We simplify the proof of earlier results by K. Schmitt and R.
Schaaf in the so-called local multiplicative case, extend them to the case of a
non-local dependence on the bifurcation parameter and to the additive case,
both in local and non-local settings.Comment: Annales Henri Poincar\'e (2009) to appea
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