We consider the problem of minimizing the $L^\infty$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^\infty$ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimizer, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.

In this paper, we study the singular boundary value problem

$$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$. We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$. Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption

\[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \]

$(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$

$m>1$

$q\in (0,\,1)$

$\sigma =\sigma _c:=2(1-q)/ (m-1)$

\[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \]

$\beta ^*\in (0,\,\infty )$

$(u_A)_{A>0}$

$(f_A)_{A>0}$

$f_A(0)=A$

$f_A'(0)=0$

$\sigma \in (0,\,\sigma _c)$

We formulate haptotaxis models of cancer invasion wherein the infiltrating cancer cells can occupy a spectrum of states in phenotype space, ranging from ‘fully mesenchymal’ to ‘fully epithelial’. The more mesenchymal cells are those that display stronger haptotaxis responses and have greater capacity to modify the extracellular matrix (ECM) through enhanced secretion of matrix-degrading enzymes (MDEs). However, as a trade-off, they have lower proliferative capacity than the more epithelial cells. The framework is multiscale in that we start with an individual-based model that tracks the dynamics of single cells, which is based on a branching random walk over a lattice representing both physical and phenotype space. We formally derive the corresponding continuum model, which takes the form of a coupled system comprising a partial integro-differential equation for the local cell population density function, a partial differential equation for the MDE concentration and an infinite-dimensional ordinary differential equation for the ECM density. Despite the intricacy of the model, we show, through formal asymptotic techniques, that for certain parameter regimes it is possible to carry out a detailed travelling wave analysis and obtain invading fronts with spatial structuring of phenotypes. Precisely, the most mesenchymal cells dominate the leading edge of the invasion wave and the most epithelial (and most proliferative) dominate the rear, representing a bulk tumour population. As such, the model recapitulates similar observations into a front to back structuring of invasion waves into leader-type and follower-type cells, witnessed in an increasing number of experimental studies over recent years.

We generalize the one-dimensional population model of Anguige & Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller & Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.

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.

In this work, we carry out an analytical and numerical investigation of travelling waves representing arced vegetation patterns on sloped terrains. These patterns are reported to appear also in ecosystems which are not water deprived; therefore, we study the hypothesis that their appearance is due to plant–soil negative feedback, namely due to biomass-(auto)toxicity interactions.

To this aim, we introduce a reaction-diffusion-advection model describing the dynamics of vegetation biomass and toxicity which includes the effect of sloped terrains on the spatial distribution of these variables. Our analytical investigation shows the absence of Turing patterns, whereas travelling waves (moving uphill in the slope direction) emerge. Investigating the corresponding dispersion relation, we provide an analytic expression for the asymptotic speed of the wave. Numerical simulations not only just confirm this analytical quantity but also reveal the impact of toxicity on the structure of the emerging travelling pattern.

Our analysis represents a further step in understanding the mechanisms behind relevant plants‘ spatial distributions observed in real life. In particular, since vegetation patterns (both stationary and transient) are known to play a crucial role in determining the underlying ecosystems’ resilience, the framework presented here allows us to better understand the emergence of such structures to a larger variety of ecological scenarios and hence improve the relative strategies to ensure the ecosystems’ resilience.

We will present the proof of existence and uniqueness of renormalized solutions to a broad family of strongly non-linear elliptic equations with lower order terms and data of low integrability. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak–Orlicz spaces. The setting considered in this paper generalized known results in the variable exponents, anisotropic polynomial, double phase and classical Orlicz setting.

We show that the energy–momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion into the orthogonal group and in particular these solutions can be nowhere $C^1$. We prove that these solutions are not stationary points if the Lagrangian $W$ is $C^1$ and strictly rank-one convex. In view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for solution of the energy–momentum equation in dimension 2, we give a sufficient condition for the non-existence of a partial $C^1$ -regularity result even under the condition that the mappings satisfy a positive Jacobian determinant condition. Finally, we consider a number of well-known functionals studied in non-linear elasticity and geometric function theory and show that these do not satisfy this obstruction to partial regularity.

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.

We make some remarks on the Euler–Lagrange equation of energy functional $I(u)=\int _\Omega f(\det Du)\,{\rm d}x,$ where $f\in C^1(\mathbb {R}).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $\Omega$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$ is not constant on $\Omega.$ We also prove some results concerning the homeomorphism solutions, non-quasimonotonicity and radial solutions, and finally we prove some stability results and discuss some related questions concerning certain approximate solutions in the 2-Dimensional cases.

For a perturbed generalized Korteweg–de Vries equation with a distributed delay, we prove the existence of both periodic and solitary waves by using the geometric singular perturbation theory and the Melnikov method. We further obtain monotonicity and boundedness of the speed of the periodic wave with respect to the total energy of the unperturbed system. Finally, we establish a relation between the wave speed and the wavelength.

In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$Laplacian operator where $1 < q < N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$, with $s\in (1/2,\,1)$, and a coercive gradient term with subcritical power $0< p<2\,s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0< p<2\,s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1< p<2$.

We consider the non-linear Schrödinger equation(Pμ)

\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*}

$\mathbb{R}^N$

$N\geq3$

$f(s)/s$

The dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis.

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$-th fractional truncated Laplacian or the $k$-th fractional eigenvalue which are fully nonlinear integral operators whose nonlocality is somehow $k$-dimensional.

We carry out the extended symmetry analysis of an ultraparabolic Fokker–Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker–Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker–Planck equation using the direct method, analyse its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalised reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker–Planck equation, in particular, those parameterised by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker–Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q<\infty$ and $2< p<\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.