A priori estimates of solutions of superlinear elliptic and parabolic problems
Július Pačuta
PhD thesis advisor: Pavol Quittner

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 [3]) and estimates of the Dirichlet heat kernel (see [1]).
We also present results from [2] on the elliptic system - Δu = a(x) |x| vq, - Δv = b(x) |x| 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.

[1] Fila M., Souplet Ph., Weissler F.B.:
Linear and nonlinear heat equations in Lδq(Ω) spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87--113.
[2] Pačuta J.:
Existence and a priori estimates for semilinear elliptic systems of Hardy type, Acta Math. Univ. Comenianae 83 (2014), 321--330.
[3] Quittner P.:
A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities, Nonlinear Analysis TM\&A 102 (2014), 144--158