Ohta examined a system of multiple speed wave equations with small initial data and demonstrated a finite time blowup. We show, in the radial case, that the same system exists almost globally with the same lifespan as a lower bound. To do this, we use integrated local energy estimates, r^p weighted local energy estimates, the Morawetz estimate that results from using the scaling vector field as a multiplier, and mixed-speed ghost weights.

In this paper, we study an optimal control problem for a viscous Cahn–Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence, and regularity are known. We show Fréchet differentiability of the associated control-to- state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.

We proved Hölder regularity for the particle trajectories of an interacting particle system in one dimension. The particle velocities were given by the nonlocal and singular interactions with the other particles. Particle collisions occur in finite time. Prior to collisions the particle velocities become unbounded, and thus the trajectories fail to be of class C¹. Our Hölder-regularity result supplements earlier studies on the well-posedness of the particle system which implies only continuity of the trajectories. Moreover, it extends and unifies several of the previously obtained estimates on the trajectories. Our proof method relied on standard ODE techniques: we transformed the system into different variables to expose and exploit the hidden monotonicity properties.

In this paper, we established the existence conditions for initial value problems involving the Caputo-Hadamard derivative operator with delay and nonlocal conditions. The problems were analyzed for various source functions. In a special case, we obtained a unique solution. These results were derived by combining the concept of the measure of non-compactness with appropriate applications of fixed point theorems. Additionally, we provided illustrative examples to support the theoretical findings.

We consider the initial boundary value problem for the compressible Navier-Stokes equations of barotropic type. It is well-known that the equations maintain its structure nearly invariant under the scaling invariance and by the Fujita–Kato principle, the problem can be solved in the scaling invariant homogeneous Besov space locally in time for the Cauchy problem. We show the initial boundary value problem of the system with 0-Dirichlet boundary condition has a unique local solution in the nearly half Euclidean space. We employ the Lagrangian transform to transfer the equation in the Lagrangian coordinate, and flattening the boundary function to the half space and show the local well-posedness of the system. The key for the proof is employing end-point L¹ maximal regularity for the parabolic equations in half space shown in [1].

We consider the isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum in a three-dimensional torus. The local well-posedness of classical solution is established by introducing a “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the velocity of the fluid near the vacuum and give some uniform estimates. In particular, the initial data allows vacuum in an open set and we do not need any initial compatibility conditions.

In this paper, we prove the existence of separable solutions to the equations of motion for self-gravitating hyperelastic matter, under an appropriate class of constitutive assumptions on the strain-energy function. Our framework includes both global-in-time solutions which expand and also solutions which collapse to a point in finite time. Other authors have constructed expanding solutions in similar settings, but to the best of our knowledge, the collapsing solutions we construct are completely new.

We propose a new continuum model for random genetic drift by employing a dynamic boundary condition approach. The model can be viewed as a regularized version of the Kimura equation and admits a continuous solution. We establish existence and uniqueness of a strong solution to the regularized system. Numerical experiments illustrate that, for sufficiently small regularization parameters, the model can capture key phenomena of the original Kimura equation, such as gene fixation and conservation of the first moment.