A priori estimates of solutions of superlinear elliptic and parabolic problems
PhD thesis advisor:
PhD thesis - Full text (548 kB)
Synopsis (316 kB)
We consider the parabolic system
ut - Δu = ur vp,
vt - Δv = uq vs, x ∈ Ω, t ∈ (0,∞),
complemented by the homogeneous Dirichlet boundary conditions and the initial
conditions (u,v)(.,0) = (u0,v0) in Ω,
where Ω is a smooth bounded domain in RN and u0,v0 ∈ 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 ) and estimates
of the Dirichlet heat kernel (see ).
We also present results from  on the elliptic system
- Δu = a(x) |x|-κ vq,
- Δv = b(x) |x|-λ up, 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.
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