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25) with the expansion coefficients C1 (t) and C2 (t) satisfying the dynamical system C1 = 2C1 C2 − σ, C2 = 2 1 + σ2 C22 . 26) Integrating the uncoupled second ODE and substituting C2 (t) = − 2(σσ+2)t into the first equation yields Kersner’s solution (1976) σ u(x, t) = A0 t − σ +2 − σ (σ +2) 2(σ +1) t− σ 2(σ +2)t x2 + , where A0 is an arbitrary constant. 25); T is the extinction time, so u(x, T ) ≡ 0. not group-invariant if A0 = 0. 24) with finite interfaces, so they describe interesting and principal phenomena of non-Darcy interface propagation with turning points, extinction patterns, quenching, etc.

33) cannot be solved explicitly, but is integrated in quadratures, giving interesting properties of finite-front propagation and evolution to traveling waves in such nonlinear media. 32). Dyson–Newman’s solution propagates for t 1 with the asymptotic speed 12 . 3. There are other applications of such solutions in the theory of reactionabsorption PDEs; see [509, p. 106] and references therein. 34) which was introduced to PDE theory in 1979 (see [245, Ch. 9] for history), plays a decisive role in blow-up combustion problems.

3. There are other applications of such solutions in the theory of reactionabsorption PDEs; see [509, p. 106] and references therein. 34) which was introduced to PDE theory in 1979 (see [245, Ch. 9] for history), plays a decisive role in blow-up combustion problems. This is the only semilinear reactiondiffusion equation of the second order that generates the regional blow-up (S-regime) for which bell-shaped solutions blow up on spatial intervals of the length 2π, [509, p. 294]. 35) where the reaction term, q(v) = v ln v, is “almost” linear as v → +∞, but, nevertheless, satisfies the Osgood criterion of blow-up, 2 ∞ ds q(s) < ∞.

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A Guided Tour of Mathematical Physics by R. Sneider


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