We study the well-posedness of solutions to the general nonlinear parabolic equations with merely integrable data in time-dependent Musielak–Orlicz spaces. With the help of a density argument, we establish the existence and uniqueness of both renormalized and entropy solutions. Moreover, we conclude that the entropy and renormalized solutions for this equation are equivalent. Our results cover a variety of problems, including those with Orlicz growth, variable exponents and double-phase growth.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in $\mathbb{R}^3$:

\begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*}

$\alpha \gt \frac {3}{4}$

$\tilde {\textrm {a}}$

$\alpha \gt \frac {1}{2}$

$\alpha \geq \frac {5}{4}$

$\frac {3}{4}\lt \alpha \lt \frac {5}{4}$

$L^2$

In this paper, we study the existence of travelling wave solutions and the spreading speed for the solutions of an age-structured epidemic model with nonlocal diffusion. Our proofs make use of the comparison principles both to construct suitable sub/super-solutions and to prove the regularity of travelling wave solutions.

We provide well-posedness results for nonlinear parabolic partial differential equations (PDEs) given by reaction–diffusion equations describing the concentration of oxygen in encapsulated cells. The cells are described in terms of a core and a shell, which introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. In addition, the cells are subject to general nonlinear consumption of oxygen. As no monotonicity condition is imposed on the consumption, monotone operator theory cannot be used. Moreover, the discontinuity in the diffusion coefficient bars us from applying classical results on strong solutions. However, by directly applying a Galerkin method, we obtain uniqueness and existence of the strong form solution. These results provide the basis to study the dynamics of cells in critical states.

This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on ${\mathbb R}^n$.

Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.

In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\equiv \mbox {const.}$, we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak’s $L^p$-Minkowski problem on $S^n$ which were obtained by other methods.

We consider a parabolic-parabolic chemotaxis system with singular chemotactic sensitivity and source functions, which is originally introduced by Short et al to model the spatio-temporal behaviour of urban criminal activities with the particular value of the chemotactic sensitivity parameter $\chi =2$. The available analytical findings for this urban crime model including $\chi =2$ are restricted either to one-dimensional setting, or to initial data and source functions with appropriate smallness, or to initial data and source functions with some radial symmetry. In the present work, our first result asserts that for any $\chi \gt 0$ the initial-boundary value problem of this urban crime model possesses a global generalised solution in the two-dimensional setting, without imposing any small or radial conditions on initial data and source functions. Our second result presents the asymptotic behaviour of such solution, under some additional assumptions on source functions.

In this article, we are concerned with the tempered fractional parabolic problem

$$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$

where $-\left (\Delta +\lambda \right )^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators.

In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$, and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$). We show the existence of two critical values $0< L_*\leq L^*$, and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. Based on the properties of $L_*$, we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$), the protection zone should start from somewhere near $0$; while if $b$ is small (i.e. $b< 1/\sqrt {-g'(0)}$), then the protection zone should start from somewhere away from $0$, and as far away from $0$ as possible.

We study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa–Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.

This paper investigates the stability of a fully parabolic–parabolic-fluid (PP-fluid) system of the Keller–Segel–Navier–Stokes type in a bounded planar domain under the natural volume-filling hypothesis. In the limit of fast signal diffusion, we first show that the global classical solutions of the PP-fluid system will converge to the solution of the corresponding parabolic–elliptic-fluid (PE-fluid) system. As a by-product, we obtain the global well-posedness of the PE-fluid system for general large initial data. We also establish some new exponential time decay estimates for suitable small initial cell mass, which in particular ensure an improvement of convergence rate on time. To further explore the stability property, we carry out three numerical examples of different types: the nontrivial and trivial equilibriums, and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.

We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:

\begin{eqnarray*}\left\{ \begin{array}{ll}u_t = u_{xx} - \chi \big(\frac{u}{v} \partial_x v \big)_x - uv +B_1(x,t),\qquad & x\in \Omega, \ t>0, \\[1mm]v_t = v_{xx} +uv - v + B_2(x,t),\qquad & x\in \Omega, \ t>0,\end{array} \right.\end{eqnarray*}

$\chi=2$

In bounded intervals $\Omega\subset\mathbb{R}$ and with prescribed suitably regular non-negative functions $B_1$ and $B_2$, we first prove the existence of global classical solutions for any choice of $\chi>0$ and all reasonably regular non-negative initial data.

We next address the issue of determining the qualitative behaviour of solutions under appropriate assumptions on the asymptotic properties of $B_1$ and $B_2$. Indeed, for arbitrary $\chi>0$, we obtain boundedness of the solutions given strict positivity of the average of $B_2$ over the domain; moreover, it is seen that imposing a mild decay assumption on $B_1$ implies that u must decay to zero in the long-term limit. Our final result, valid for all $\chi\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ which contains the relevant value $\chi=2$, states that under the above decay assumption on $B_1$, if furthermore $B_2$ appropriately stabilises to a non-trivial function $B_{2,\infty}$, then (u,v) approaches the limit $(0,v_\infty)$, where $v_\infty$ denotes the solution of

\begin{eqnarray*}\left\{ \begin{array}{l}-\partial_{xx}v_\infty + v_\infty = B_{2,\infty},\qquad x\in \Omega, \\[1mm]\partial_x v_{\infty}=0,\qquad x\in\partial\Omega.\end{array} \right.\end{eqnarray*}

$\chi$

$\chi$

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system, we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularising terms.

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

In this paper, we investigate the fast signal diffusion limit of solutions of the fully parabolic Keller–Segel–Stokes system to solution of the parabolic–elliptic-fluid counterpart in a two-dimensional or three-dimensional bounded domain with smooth boundary. Under the natural volume-filling assumption, we establish an algebraic convergence rate of the fast signal diffusion limit for general large initial data by developing a series of subtle bootstrap arguments for combinational functionals and using some maximal regularities. In our current setting, in particular, we can remove the restriction to asserting convergence only along some subsequence in Wang–Winkler and the second author (Cal. Var., 2019).