One-sided Lipschitz inequality for scalar balance laws with time-dependent source

Key Points

• The unique entropy solution exhibits $BV$ regularity under specific conditions between the flux and the nonlinear source term.
• A generalized Oleinik-type inequality establishes a new framework for analyzing convex scalar balance laws with time-dependent sources.
• Conditions applied result in positive $BV$ regularity results, while fractional $BV$ regularity may be anticipated in alternative scenarios.
• These findings indicate potential applications in understanding the behavior of solutions in various mathematical and physical contexts.