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Download PDF by Quittner P., Souplet P.: Superlinear Parabolic Problems: Blow-up, Global Existence

By Quittner P., Souplet P.

This booklet is dedicated to the qualitative research of ideas of superlinear elliptic and parabolic partial differential equations and platforms. This category of difficulties includes, particularly, a few reaction-diffusion structures which come up in a number of mathematical types, particularly in chemistry, physics and biology.The booklet is self-contained and up to date, it has a excessive didactic caliber. it's dedicated to difficulties which are intensively studied yet haven't been taken care of to this point extensive within the publication literature. The meant viewers comprises graduate and postgraduate scholars and researchers operating within the box of partial differential equations and utilized arithmetic.

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Download e-book for kindle: Superlinear Parabolic Problems: Blow-up, Global Existence by Quittner P., Souplet P.

This publication is dedicated to the qualitative examine of recommendations of superlinear elliptic and parabolic partial differential equations and structures. This classification of difficulties includes, specifically, a few reaction-diffusion platforms which come up in quite a few mathematical types, particularly in chemistry, physics and biology.

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Extra info for Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

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Zn ) = v(−z1 , z2 , . . , zn ). Repeating this procedure with any given direction instead of e1 we see that v, hence u, are radially symmetric (about zero). If we repeat this procedure with u ˜(x) = u(x − x0 ) for a given x0 = 0 instead of u, we show that u is radially symmetric about the point x0 . Since this is true for any x0 , the function u has to be constant. 1) is the trivial solution. If p = pS and µ ¯ < 0, then v is symmetric with respect to {z1 = µ ¯}. If µ ¯ = 0, then we can repeat the procedure with v˜(z) := v(z 0 ) and in any case we obtain ˜} for suitable µ ˜.

9. (i) Positive solutions in the critical case [98]. Let Ω be bounded, p = pS , λ∗ := inf{λ ∈ (0, λ1 ) : Sλ (Ω) < S}. Set uε (x) := (ε + |x|2 )−(n−2)/2 (cf. 6) and assume 0 ∈ Ω. If n ≥ 4 and λ > 0, then careful estimates show Sλ (uε ϕ, Ω) < S provided ϕ ∈ D(Ω) is nonnegative, ϕ = 1 in a neighborhood of 0 and ε is small enough. 1) possesses a positive solution for any λ ∈ (0, λ1 ). Now let n = 3 and Ω = B1 (0). If λ > λ1 /4, then Sλ (uε ϕ, Ω) < S provided ϕ(x) = cos(π|x|/2) and ε is small enough.

1) if p ≤ pS . 3) will be found by minimizing the functional Ψ with respect to the set M := {u : Φ(u) = 1} and using the following well-known Lagrange multiplier rule. 6. 1. Let X be a real Banach space, w ∈ X and let Ψ, Φ1 , . . , Φk : X → R be C 1 in a neighborhood of w. Denote M := {u ∈ X : Φi (u) = Φi (w) for i = 1, . . , k} and assume that w is a local minimizer of Ψ with respect to the set M . If Φ1 (w), . . , Φk (w) are linearly independent, then there exist µ1 , . . , µk ∈ R such that k Ψ (w) = µi Φi (w).

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Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States by Quittner P., Souplet P.


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