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: $$ u_t + \frac{1}{2}\,\bl{a}\cdot\nabla_x u^2=\Delta u_+ \qquad \text{ for } t>0, \leqno{(*)} $$ where a is a constant vector and u+ = max{u,0}. The equation is hyperbolic in the region [u 0]. It is shown that entropy solutions to (*) that grow at most linearly as $|x|\rightarrow\infty$ are stable in a weighted $L^1(\rn)$ space, which implies that the solutions are unique. The linear growth as $|x|\rightarrow\infty$ imposed on the solutions is shown to be optimal for uniqueness to hold. The same results hold if the Burgers nonlinearity $\frac{1}{2}\,\bl{a}u^2$ is replaced by a general flux function f(u), provided f'(u(x,t)) grows in x at most linearly as $|x|\rightarrow\infty$, and/or the degenerate term u+ is replaced by a nondecreasing, degenerate, Lipschitz continuous function $\beta(u)$ defined on $\rr$. For more general $\beta(\cdot)$, the results continue to hold for bounded solutions.