Existence of extremal periodic solutions for quasilinear parabolic equations.

Carl, Siegfried

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We consider a nonlinear differential-functional parabolic boundary initial value problem (1) ⎧A z + f(x,z(t,x),z(t,·)) - ∂z/∂t = 0 for t > 0, x ∈ G, ⎨z(t,x) = h(x) for t > 0, x ∈ ∂G, ⎩z(0,x) = φ₀(x) for x ∈ G, and the associated elliptic boundary value problem with Dirichlet condition (2) ⎧Az + f(x,z(x),z(·)) = 0 for x ∈ G, ⎨z(x) = h(x) for x ∈ ∂G ⎩ where $x = (x ₁, . . ., x_{m}) \in G \subset ℝ^{m}$ , G is an open and bounded domain with $C^{2 + α}$ (0 < α ≤ 1) boundary, the operator Az := ∑j,k=1m ajk(x) (∂²z/(∂xj...

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We review some recent results concerning a priori bounds for solutions of superlinear parabolic problems and their applications.