1,775 research outputs found
Well-posedness for a class of nonlinear degenerate parabolic equations
In this paper we obtain well-posedness for a class of semilinear weakly
degenerate reaction-diffusion systems with Robin boundary conditions. This
result is obtained through a Gagliardo-Nirenberg interpolation inequality and
some embedding results for weighted Sobolev spaces
Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states
In this paper we study the global approximate multiplicative controllability
for nonlinear degenerate parabolic Cauchy problems. In particular, we consider
a one-dimensional semilinear degenerate reaction-diffusion equation in
divergence form governed via the coefficient of the \-reaction term (bilinear
or multiplicative control). The above one-dimensional equation is degenerate
since the diffusion coefficient is positive on the interior of the spatial
domain and vanishes at the boundary points. Furthermore, two different kinds of
degenerate diffusion coefficient are distinguished and studied in this paper:
the weakly degenerate case, that is, if the reciprocal of the diffusion
coefficient is summable, and the strongly degenerate case, that is, if that
reciprocal isn't summable. In our main result we show that the above systems
can be steered from an initial continuous state that admits a finite number of
points of sign change to a target state with the same number of changes of sign
in the same order. Our method uses a recent technique introduced for uniformly
parabolic equations employing the shifting of the points of sign change by
making use of a finite sequence of initial-value pure diffusion pro\-blems. Our
interest in degenerate reaction-diffusion equations is motivated by the study
of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers
model) and some models in population genetics (see, e.g., the Fleming-Viot
model).Comment: arXiv admin note: text overlap with arXiv:1510.0420
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
We consider a quasilinear PDE system which models nonlinear vibrations of a
thermoelastic plate defined on a bounded domain in R^n. Well-posedness of
solutions reconstructing maximal parabolic regularity in nonlinear
thermoelastic plates is established. In addition, exponential decay rates for
strong solutions are also shown.Comment: 14 page
Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative
This paper investigates the regional gradient controllability for ultra-slow
diffusion processes governed by the time fractional diffusion systems with a
Hadamard-Caputo time fractional derivative. Some necessary and sufficient
conditions on regional gradient exact and approximate controllability are first
given and proved in detail. Secondly, we propose an approach on how to
calculate the minimum number of strategic actuators. Moreover, the
existence, uniqueness and the concrete form of the optimal controller for the
system under consideration are presented by employing the Hilbert Uniqueness
Method (HUM) among all the admissible ones. Finally, we illustrate our results
by an interesting example.Comment: 16 page
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