2,475 research outputs found
Nonlinear parabolic problems in Musielak--Orlicz spaces
Our studies are directed to the existence of weak solutions to a parabolic
problem containing a multi-valued term. The problem is formulated in the
language of maximal monotone graphs. We assume that the growth and coercivity
conditions of a nonlinear term are prescribed by means of time and space
dependent --function. This results in formulation of the problem in
generalized Musielak-Orlicz spaces. We are using density arguments, hence an
important step of the proof is a uniform boundedness of appropriate convolution
operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a
kind of logarithmic H\"older regularity with respect to and .Comment: 33 page
A doubly nonlinear evolution problem related to a model for microwave heating
This paper is concerned with the existence and uniqueness of the solution to
a doubly nonlinear parabolic problem which arises directly from a circuit model
of microwave heating. Beyond the relevance from a physical point of view, the
problem is very interesting also in a mathematical approach: in fact, it
consists of a nonlinear partial differential equation with a further
nonlinearity in the boundary condition. Actually, we are going to prove a
general result: the two nonlinearities are allowed to be maximal monotone
operators and then an existence result will be shown for the resulting problem.Comment: Key words and phrases: nonlinear parabolic equation, nonlinear
boundary condition, existence of solution
On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition
We study a coupled bulk-surface Allen-Cahn system with an affine linear
transmission condition, that is, the trace values of the bulk variable and the
values of the surface variable are connected via an affine relation, and this
serves to generalize the usual dynamic boundary conditions. We tackle the
problem of well-posedness via a penalization method using Robin boundary
conditions. In particular, for the relaxation problem, the strong
well-posedness and long-time behavior of solutions can be shown for more
general and possibly nonlinear relations. New difficulties arise since the
surface variable is no longer the trace of the bulk variable, and uniform
estimates in the relaxation parameter are scarce. Nevertheless, weak
convergence to the original problem can be shown. Using the approach of Colli
and Fukao (Math. Models Appl. Sci. 2015), we show strong existence to the
original problem with affine linear relations, and derive an error estimate
between solutions to the relaxed and original problems.Comment: 34 page
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