Home Calculus • New PDF release: Blow-up Theories for Semilinear Parabolic Equations

New PDF release: Blow-up Theories for Semilinear Parabolic Equations

By Bei Hu

There is a gigantic quantity of labor within the literature in regards to the blow-up habit of evolution equations. it's our goal to introduce the speculation by way of emphasizing the equipment whereas trying to steer clear of tremendous technical computations. to arrive this target, we use the easiest equation to demonstrate the tools; those tools quite often practice to extra common equations.

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30], or [20, p. 1]. 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.

34) 2 , The right-hand side of the above inequality is unbounded as T → ∞ if p 1 + np 2 2 which gives a contradiction in this case. In the case 1 + np < p < 1 + n , we have n(p−1)2 2 > −1 + (p−1)np , so we get a contradiction by letting T → ∞ and then 2 taking t 1. 29), for t > 1, up (x, t) up1 (x, t) cp1 p|x|2 . 35) Substituting this estimate into the expression for u2 , we obtain, for t > 2, u(x, t) u2 (x, t) t 1 Rn c4 tn/2 × c4 tn/2 × = cp1 p|y|2 dydτ exp − 2τ (2πτ )np/2 t/2 tn/2 τ 1+n/2 (t − τ )n/2 1 |x|2 + |y|2 p|y|2 dy dτ − − 2(t − τ ) 2τ t/2 tn/2 1+n/2 τ (t − τ )n/2 1 2 2 |x| + |y| p|y|2 dy dτ − − t 2τ t/2 dτ τ 1 Γ (x − y, t − τ ) exp − Rn exp exp − Rn exp c5 n/2 t exp − c5 n/2 t exp − |x|2 t |x|2 t |x|2 t |x|2 t |x|2 t |x|2 log(t/2).

4, we have −f (0)(n(x0 )·γ)2 > 0. By continuity 2 ∂ u ∂γ 2 (x) ∂2 u ∂γ 2 (x0 ) = Δu(x0 )(n(x0 ) · γ)2 = > 0 on Ω ∩ Bδ (x0 ) if δ > 0 is small.

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