This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

We give a Hopf boundary point lemma for weak solutions of linear divergence form uniformly elliptic equations, with Hölder continuous top-order coefficients and lower-order coefficients in a Morrey space.

We use bifurcation and topological methods to investigate the existence/nonexistence and the multiplicity of positive solutions of the following quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\kappa \Delta \left( {u^2} \right)u = \beta u-\lambda \Phi \left( {u^2} \right)u{\mkern 1mu} {\mkern 1mu} } \hfill & {{\rm in}\;\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm on}\;\partial \Omega } \hfill \cr } } \right.$$

In this paper, we analyse the semilinear fourth-order problem ( − Δ)2u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and

$$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$

In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity

$$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$

We consider an elliptic problem of the type

$$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$

$\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $

Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.

As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem

$$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$

Let n ⩾ 3, 0 ⩽ m < n − 2/n, ρ1 > 0, $\beta>\beta_{0}^{(m)}=(({m\rho_{1}})/({n-2-nm}))$, αm = ((2β + ρ1)/(1 − m)) and α = 2β+ρ1. For any λ > 0, we prove the uniqueness of radially symmetric solution υ(m) of Δ(υm/m) + αmυ + βx · ∇υ = 0, υ > 0, in ℝn∖{0} which satisfies $\lim\nolimits_{|x|\to 0|}|x|^{\alpha _m/\beta }v^{(m)}(x) = \lambda ^{-((\rho _1)/((1-m)\beta ))}$ and obtain higher order estimates of υ(m) near the blow-up point x = 0. We prove that as m → 0+, υ(m) converges uniformly in C2(K) for any compact subset K of ℝn∖{0} to the solution υ of Δlog υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn\{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$. We also prove that if the solution u(m) of ut = Δ (um/m), u > 0, in (ℝn∖{0}) × (0, T) which blows up near {0} × (0, T) at the rate $ \vert x \vert ^{-{\alpha_{m}}/{\beta}}$ satisfies some mild growth condition on (ℝn∖{0}) × (0, T), then as m → 0+, u(m) converges uniformly in C2 + θ, 1 + θ/2(K) for some constant θ ∈ (0, 1) and any compact subset K of (ℝn∖{0}) × (0, T) to the solution of ut = Δlog u, u > 0, in (ℝn∖{0}) × (0, T). As a consequence of the proof, we obtain existence of a unique radially symmetric solution υ(0) of Δ log υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn∖{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$.

We prove a Carleman estimate for elliptic second-order partial differential expressions with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function u ∈ W2,2 with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular, we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and Lipschitz constant.

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ > 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.

Pullback attractors with forwards unbounded behaviour are to be found in the literature, but not much is known about pullback attractors with each and every section being unbounded. In this paper, we introduce the concept of unbounded pullback attractor, for which the sections are not required to be compact. These objects are addressed in this paper in the context of a class of non-autonomous semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as the initial time goes to -∞, for each elapsed time. Distinct regimes for the non-autonomous term are taken into account. Namely, we address the small non-autonomous perturbation and the asymptotically autonomous cases.

In the present paper, we consider the nonlocal Kirchhoff problem

$$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$

$\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$

In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:

$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$

$M:{\open R}_0^ + \to {\open R}^ + $

We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

We consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_{p}(\unicode[STIX]{x1D6FA})$, where $\unicode[STIX]{x1D6FA}=\mathbf{R}^{d}\backslash K$ with $K$ a closed convex subset of $\mathbf{R}^{d}$. Let $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ denote the boundary of $\unicode[STIX]{x1D6FA}$ and $d_{\unicode[STIX]{x1D6E4}}$ the Euclidean distance to $\unicode[STIX]{x1D6E4}$. We consider weighting functions $c_{\unicode[STIX]{x1D6FA}}=c\circ d_{\unicode[STIX]{x1D6E4}}$ with $c(s)=s^{\unicode[STIX]{x1D6FF}}(1+s)^{\unicode[STIX]{x1D6FF}^{\prime }-\unicode[STIX]{x1D6FF}}$ and $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$. Then the Hardy inequalities take the form

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D711}|^{p}\geq b_{p}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-p}|\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}|H\unicode[STIX]{x1D711}|^{p}\geq d_{p}\int _{\unicode[STIX]{x1D6FA}}|c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-2}\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$

$H=-\text{div}(c_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FB})$

$b_{p},d_{p}$

$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$

We study the initial boundary value problem for a fourth-order parabolic equation with nonstandard growth conditions. We establish the local existence of weak solutions and derive the finite time blow-up of solutions with nonpositive initial energy.

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic–poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

We introduce an Almgren frequency function of the sub-$p$-Laplace equation on the Heisenberg group to establish a doubling estimate under the assumption that the frequency function is locally bounded. From this, we obtain some partial results on unique continuation for the sub-$p$-Laplace equation.

The Wasserstein gradient flow structure of the partial differential equation system governing multiphase flows in porous media was recently highlighted in Cancès et al. [Anal. PDE10(8), 1845–1876]. The model can thus be approximated by means of the minimising movement (or JKO after Jordan, Kinderlehrer and Otto [SIAM J. Math. Anal.29(1), 1–17]) scheme that we solve thanks to the ALG2-JKO scheme proposed in Benamou et al. [ESAIM Proc. Surv.57, 1–17]. The numerical results are compared to a classical upstream mobility finite volume scheme, for which strong stability properties can be established.