Blow-up Theories for Semilinear Parabolic Equations by Bei Hu

By Bei Hu

There is a gigantic volume of labor within the literature concerning the blow-up habit of evolution equations. it's our goal to introduce the idea through emphasizing the equipment whereas trying to steer clear of great technical computations. to arrive this objective, we use the easiest equation to demonstrate the equipment; those tools quite often follow to extra normal equations.

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3. The combined boundary-interior estimates (without initial condition) are also valid. 12) for x ∈ Ω. 5 (Existence and uniqueness, Dirichlet). [94, p. 14]. Let ∂Ω ∈ C 2+α (0 < α < 1). 7), f ∈ C α,α/2 (Ω T ), g ∈ C 2+α,1+α/2 (Ω T ), and u0 ∈ C 2+α (Ω) satisfies the second order compatibility conditions. 13) admits a unique solution u ∈ C 2+α,1+α/2 (Ω T ). 6. [94, p. 6]. 8) and c const . 14) 0. 15) for any direction η such that η · n > 0, where n is the exterior normal vector. 4. ΩT may be replaced by a general set Q ⊂ Rn × [0, T ] which satisfies the interior ellipsoid condition (see [94]).

Assume further that c c0 > 0. Prove that there exists a unique periodic solution u ∈ C 2+α,1+α/2 (Ω × (−∞, ∞)) to the parabolic equation with period T . • Chapter 4 A Review of Fixed Point Theorems We collect in this chapter several fixed point theorems, which are useful for proving existence of solutions to nonlinear equations and systems. 1 (Contraction mapping principle). ([67, p. 1], see also the note after the Theorem for replacing the whole space by a closed subset). Let X be a Banach space and let K be a closed convex set in X.

In addition to the assumptions in the above theorem, we assume ∂Ω satisfies a uniform exterior cone condition and that u is Hölder continuous on the lateral boundary and on the initial manifold and zeroth order compatibility condition on ∂Ω × {t = 0}, then the above estimates extend to Ω T . The Hölder estimates for the case of oblique boundary condition can be found in [94, p. 36]. 7 Embedding Theorems The parabolic version of the embedding theorems is very similar to the elliptic version when the t derivatives is considered “half the order” of x-derivatives.

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