Abstract
We consider the equation u t =(ϕ(u) ψ (u x )) x , where ϕ>0 and where ψ is a strictly increasing function with lim s→∞ ψ=ψ ∞<∞. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u t =ψ ∞ (ϕ(u)) x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.
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Bertsch, M., Dal Passo, R. Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch. Rational Mech. Anal. 117, 349–387 (1992). https://doi.org/10.1007/BF00376188
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DOI: https://doi.org/10.1007/BF00376188