Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations

Consider the Cauchy problem for the nonlinear hyperbolic-parabolic equation: ut + 1/2a · ∇xu2 = Δu+ for t > 0, where a is a constant vector and u+ = max{u, 0}. The equation is hyperbolic in the region [u < 0] and parabolic in the region [u > 0]. It is shown that entropy solutions to (*) that grow at most linearly as |x| → ∞ are stable in a weighted L1(ℝN) space, which implies that the solutions are unique. The linear growth as |x| → ∞ imposed on the solutions is shown to be optimal for uniqueness to hold. The same results hold if the Burgers nonlinearity 1/2 au2 is replaced by a general flux function f(u), provided f′(u(x, t)) grows in x at most linearly as |x| → ∞, and/or the degenerate term u+ is replaced by a nondecreasing, degenerate, Lipschitz continuous function β(u) defined on ℝ. For more general β(·), the results continue to hold for bounded solutions.