SynopsisA fundamental comparison theorem is established for general semilinear parabolic systems via the notions of sectorial operators, analytic semigroups and the application of the Tychonoff Fixed Point Theorem. Based on this result, we establish a maximum principle for systems of general parabolic operators and general comparison theorems for parabolic systems with quasimonotone or mixed quasimonotone nonlinearities. These results cover and extend most currently used forms of maximum principles and comparison theorems. A global existence theorem for parabolic systems is derived as an application which, in particular, gives rise to some global existence results for Fujita type systems and certain generalisations.
AbstractWe consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$
C
(
∂
M
)
of continuous functions on the boundary $$\partial M$$
∂
M
of a compact manifold $$\overline{M}$$
M
¯
with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$
π
2
, generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$
π
2
on the space $$\mathrm {C}(\overline{M})$$
C
(
M
¯
)
.
AbstractWe consider systems of parabolic equations coupled in zero and first order terms.
We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain.
The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.
AbstractWe are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $$L_{q,r}(0,T;L_p(\Omega ))$$
L
q
,
r
(
0
,
T
;
L
p
(
Ω
)
)
spaces.
Switching Controls for Analytic Semigroups and Applications to Parabolic Systems
