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 2 a · ∇xu = ∆u+ 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 L(IR ) 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 12 au 2 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 non-decreasing, degenerate, Lipschitz continuous function β(u) defined on IR. For more general β(·), the results continue to hold for bounded solutions. 1 Department of Mathematics, Northwestern University, Evanston, IL 60208–2730; Email: [email protected]; Phone: 847-491-5553; Fax: 847-491-8906; Partially supported by NSF Grants DMS-9971793, INT-9987378, and INT-9726215. 2 Department of Mathematics, Vanderbilt University, Nashville, TN 60208–2730; Email: [email protected]; Phone: 615-343-5906; Fax: 615-343-0215; Partially supported by NSF Grants DMS-9706388 and DMS-9708261.