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The paper establishes the existence and long time behavior of BV solutions to a hyperbolic system of balance laws with partially dissipative source, manifesting relaxation. The component of the state vector that is actively damped decays to zero, while the complementary component, whose total mass is not necessarily conserved, behaves asymptotically as a heat kernel. [ABSTRACT FROM AUTHOR]

The rate of convergence for vanishing viscosity approximations to hyperbolic balance laws is established. The result applies to systems that are strictly hyperbolic and genuinely nonlinear with a source term satisfying a special mechanism that induces dissipation. The proof relies on error estimates that measure the interaction of waves. Shock waves are treated by monitoring the evolution of suitable Lyapunov functionals, whereas interactions involving rarefaction waves are accommodated by employing a sharp decay estimate. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for an n x n strictly hyperbolic system of balance laws ut + f(u)x = g(x, u), x ∈ ℝ, t > 0, ||g(x, ·)||C² ≤ M̃ (x) ∈ L¹, endowed with the initial data u(0, .) = uo ∈ L¹ ∩ BV(ℝ; ℝn). Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., |λi(u)| ≥ c > 0 for all i ∈ {1, . . . ,n}. Assuming that the L¹ norms of ||g(x, ·)||C¹ and ||uo||BV(ℝ) are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D. Amadori, L. Gosse, and G. Guerra, Arch. Ration. Mech. Anal., 162 (2002), pp. 327-366] to unbounded (in L∞) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup. [ABSTRACT FROM AUTHOR]

We investigate viscous shock profiles of the Riemann problem for systems of hyperbolic balance laws. Even strictly hyperbolic flux terms together with a nonoscillating kinetic part can lead to oscillating viscous shock profiles. They appear near a Hopf-like bifurcation point of the traveling wave equation. [ABSTRACT FROM AUTHOR]