In this paper, we investigate a wear and mixed shock model in which the system can fail due to internal aging or external shocks. The lifetime of the system, due to internal wear, follows continuous phase-type (PH) distributions. The external random shocks arrive at the system according to a PH renewal process. The system will fail when the internal failure occurs or $k_1$ consecutive external shocks, the size of at least $d_1$ or $k_2$ consecutive external shocks the size of at least $d_2$ occur, where $d_1<d_2$, $k_1>k_2$. The failed system can be repaired immediately, and the repair times of the system are governed by continuous PH distributions. The system can be replaced by a new and identical one based on a bivariate replacement policy $(L,N)$. The long-run average profit rate for the system is obtained by employing the closure property of the PH distribution. Finally, a numerical example is also given to determine the optimal replacement policy.

This article concerns the regularity of weak solutions for a variational inequality problem constructed by a fourth-order parabolic operator which has received much attention recently. We first consider the internal regular estimate of weak solutions using the difference type test function. Then, the near edge regularity and global regularity of weak solutions are analyzed by using the finite cover principle. Since the quadratic gradient of the weak solution does not satisfy the conditions for a test function, we have constructed a test function using a spatial difference operator to complete the proof of regularity. The results show that the weak solution has a second order regularity and an $ {L^\infty }(0, T;{H^2}(\Omega)) $ estimation independent of the lower order norm of the weak one.

Systems that perform critical missions will often be affected by internal degradation until they reach a failure state. For safety critical systems that perform certain tasks, failure of the systems will have serious consequences. In such scenarios, the survival of the systems has a higher priority than the completion of the task. The task can be suspended at an appropriate time and a rescue procedure can be initiated to reduce the risk of system failures. When the task is important, the systems can try to execute the task multiple times after the task is abandoned and the rescue is completed, to improve the probability of task completion. This study further expands the existing research on multiple task abandon strategy by proposing degradation‐based multicriteria mission abandon policies considering multiattempts and two types of task success criteria. The task is abandoned dynamically based on the degradation level and time in the mission in each attempt. Under the dynamic abandon policies, mission reliability, and systems survivability are evaluated using the recursive method. The optimal abandon thresholds are investigated numerically.