We consider the discrete Safronov-Dubovskiĭ aggregation equation associated with the physical condition, where particle injection and extraction take place in the dynamical system. In application, this model is used to describe the aggregation of particle-monomers in combination with sedimentation of particle-clusters. More precisely, we prove well-posedness of the considered model for a large class of aggregation kernel with source and efflux coefficients. Furthermore, over a long time period, we prove that the dynamical model attains a unique equilibrium solution with an exponential rate under a suitable condition on the forcing coefficient.

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

In this work we study the Cauchy problem in Gevrey spaces for a generalized class of equations that contains the case $b=0$ of the $b$-equation. For the generalized equation, we prove that it is locally well-posed for initial data in Gevrey spaces. Moreover, as we move to global well-posedness, we show that for a particular choice of the parameter in the equation the local solution is global analytic in both time and spatial variables.

A system of mutually interacting superprocesses with migration is constructed as the limit of a sequence of branching particle systems arising from population models. The uniqueness in law of the superprocesses is established using the pathwise uniqueness of a system of stochastic partial differential equations, which is satisfied by the corresponding system of distribution function-valued processes.

When a liquid fills the semi-infinite space between two concentric cylinders which rotate at different steady speeds, how about the shape of the free surface on top of the fluid? The different fluids will lead to a different shape. For the Newtonian fluid, the meniscus descends due to the centrifugal forces. However, for the certain non-Newtonian fluid, the meniscus climbs the internal cylinder. We want to explain the above phenomenon by a rigorous mathematical analysis theory. In the present paper, as the first step, we focus on the Newtonian fluid. This is a steady free boundary problem. We aim to establish the well-posedness of this problem. Furthermore, we prove the convergence of the formal perturbation series obtained by Joseph and Fosdick in Arch. Ration. Mech. Anal. 49 (1973), 321–380.

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces ${{C}^{\alpha }}(\mathbb{R};\,X)$, we completely characterize the ${{C}^{\alpha }}$-well-posedness of the first order degenerate differential equations with finite delay $(Mu{)}'(t)\,=\,Au(t)\,+\,F{{u}_{t}}\,+\,f(t)$ for $t\,\in \,\mathbb{R}$ by the boundedness of the $(M,\,F)$-resolvent of A under suitable assumption on the delay operator $F$, where $A,M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\,\cap \,D(M)\,\ne \,\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r,0];X)$ to $X$, and $r\,>\,0$ is fixed.

A fully discrete A-ϕ finite element scheme for a nonlinear model of type-II superconductors is proposed and analyzed. The nonlinearity is due to a field dependent conductivity with the regularized power-law form. The challenge of this model is the error estimate for the nonlinear term under the time derivative. Applying the backward Euler method in time discretisation, the well-posedness of the approximation problem is given based on the theory of monotone operators. The fully discrete system is derived by standard finite element method. The error estimate is suboptimal in time and space.

We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

A well-posedness theory for the initial-value problem for hydroelastic waves in two spatial dimensions is presented. This problem, which arises in numerous applications, describes the evolution of a thin elastic membrane in a two-dimensional (2D) potential flow. We use a model for the elastic sheet that accounts for bending stresses and membrane tension, but which neglects the mass of the membrane. The analysis is based on a vortex sheet formulation and, following earlier analyses and numerical computations in 2D interfacial flow with surface tension, we use an angle–arclength representation of the problem. We prove short-time well-posedness in Sobolev spaces. The proof is based on energy estimates, and the main challenge is to find a definition of the energy and estimates on high-order non-local terms so that an a priori bound can be obtained.

We analyse a nonlinear hierarchical size-structured population model with time-dependent individual vital rates. The existence and uniqueness of nonnegative solutions to the model are shown via a comparison principle. Our investigation extends some results in the literature.

In this paper, we first discuss the well-posedness of linearizing equations, and then study the stability and unstability of the 3-D compressible Euler Equation, by analysing the existence of saddle point. In addition, we give the existence of local solutions of the compressible Euler equation.

We consider a non-local version of the Cahn–Hilliard equation characterized by the presence of a fractional diffusion operator, and which is subject to fractional dynamic boundary conditions. Our system generalizes the classical system in which the dynamic boundary condition was used to describe any relaxation dynamics of the order-parameter at the walls. The proposed fractional dynamic boundary condition appears to be more general in the sense that it incorporates non-local effects which were completely ignored in the classical approach. We aim to deduce well-posedness and regularity results as well as to establish the existence of finite-dimensional attractors for this system.

We study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

We consider first-order hyperbolic systems on an interval with dynamic boundary conditions. These systems occur when the ordinary differential equation dynamics on the boundary interact with the waves in the interior. The well-posedness for linear systems is established using an abstract Friedrichs theorem. Due to the limited regularity of the coefficients, we need to introduce the appropriate space of test functions for the weak formulation. It is shown that the weak solutions exhibit a hidden regularity at the boundary as well as at interior points. As a consequence, the dynamics of the boundary components satisfy an additional regularity. Neither result can be achieved from standard semigroup methods. Nevertheless, we show that our weak solutions and the semigroup solutions coincide. For illustration, we give three particular physical examples that fit into our framework.

We consider a diffuse interface model for tumour growth consisting of a Cahn–Hilliard equation with source terms coupled to a reaction–diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn–Hilliard equation leads to new difficulties when one aims to derive a priori estimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol.67 1457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.