PhD thesis advisor:

PhD thesis - Full text (548 kB)

Synopsis (316 kB)

We consider the parabolic system
*u*_{t} - Δu = u^{r} v^{p},
*v*_{t} - Δv = u^{q} v^{s}, *x ∈ Ω*, *t ∈ (0,∞)*,
complemented by the homogeneous Dirichlet boundary conditions and the initial
conditions *(u,v)(.,0) = (u*_{0},v_{0}) in *Ω*,
where *Ω* is a smooth bounded domain in *R*^{N} and *u*_{0},v_{0} ∈ L^{∞}(Ω)
are nonnegative functions.
We find conditions on *p, q, r, s* guaranteeing a priori estimates of nonnegative
classical global solutions. More precisely every such solution is bounded
by a constant depending on suitable norm of the initial data.
Our proofs are based on bootstrap in weighted Lebesgue spaces,
universal estimates of auxiliary functions (see [3]) and estimates
of the Dirichlet heat kernel (see [1]).

We also present results from [2] on the elliptic system*- Δu = a(x) |x|*^{-κ} v^{q},
*- Δv = b(x) |x|*^{-λ} u^{p}, *x ∈ Ω*,
complemented by the homogeneous Dirichlet boundary condition,
where *Ω* is a smooth bounded domain with the origin lying on its boundary, *a,b ∈ L*^{∞}(Ω)
are nonnegative and not identically zero.
Under some assumptions on *p, q, κ, λ*
we prove a priori estimates and existence of positive very weak solutions
of the system.

We also present results from [2] on the elliptic system

[1] Fila M., Souplet Ph., Weissler F.B.:

[2] Pačuta J.:

[3] Quittner P.: