Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of \(c\ge c^*\) for the degenerate reaction–diffusion equation without delay, where \(c^*>0\) is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay \(\tau >0\). Furthermore, we prove the global existence and uniqueness of \(C^{\alpha ,\beta }\)-solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted \(L^1\)-space. The exponential convergence rate is also derived.     

Existence and Uniqueness of Invariant Measures for Stochastic Reaction–Diffusion Equations in Unbounded Domains

In this paper, we investigate the long-time behavior of stochastic reaction–diffusion equations of the type \(\text {d}u = (Au + f(u))\text {d}t + \sigma (u) \text {d}W(t)\), where \(A\) is an elliptic operator, \(f\) and \(\sigma \) are nonlinear maps and \(W\) is an infinite-dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function \(f\) possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure, which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper, we expand the existing classes of nonlinear functions \(f\) and \(\sigma \) and elliptic operators \(A\) for which the invariant measure exists, in particular in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if \(A\) is the Shrödinger-type operator \(A = \frac{1}{\rho }(\text {div} \rho \nabla u)\) where \(\rho = \text {e}^{-|x|^2}\) is the Gaussian weight.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.     

A potential well for an integral functional with singular Lagrangian

This paper concerns a functional of the form

$$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$

on the Sobolev space \(H_0^1(\Omega )\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N\) with \(N\ge 3\) and \(0\in \Omega \). The hypotheses on L ensure that \(u\equiv 0\) is a critical point of \(\Phi \), but allow the Lagrangian to be singular at \(x=0\). It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of \(\Phi \) at \(u\equiv 0\)), Legendre (ellipticity at \(u\equiv 0\)) and Weierstrass [strict convexity of \(L(x,s,\xi )\) with respect to \(\xi \)] from the calculus of variations are not sufficient ensure that \(u\equiv 0\) is a local minimum of \(\Phi \). Using recent criteria for the existence of a potential well of a \(C^1\)-functional on a real Hilbert space, conditions implying that \(u\equiv 0\) lies in a potential well of \(\Phi \) are established. They are shown to be sharp in some cases.

     

Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form \(-\Delta _p u = |\nabla u|^p + \sigma \) in a bounded domain \(\Omega \subset \mathbb {R}^n\). Here \(\Delta _p\), \(p>1\), is the standard p-Laplacian operator defined by \(\Delta _p u=\mathrm{div}\, (|\nabla u|^{p-2}\nabla u)\), and the datum \(\sigma \) is a signed distribution in \(\Omega \). The class of solutions that we are interested in consists of functions \(u\in W^{1,p}_0(\Omega )\) such that \(|\nabla u|\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\), a space pointwise Sobolev multipliers consisting of functions \(f\in L^{p}(\Omega )\) such that

$$\begin{aligned} \int _{\Omega } |f|^{p} |\varphi |^p dx \le C \int _{\Omega } (|\nabla \varphi |^p + |\varphi |^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$

for some \(C>0\). This is a natural class of solutions at least when the distribution \(\sigma \) is nonnegative and compactly supported in \(\Omega \). We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write \(\sigma =\mathrm{div}\, F\) for a vector field F such that \(|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\). As an important application, via the exponential transformation \(u\mapsto v=e^{\frac{u}{p-1}}\), we obtain an existence result for the quasilinear equation of Schrödinger type \(-\Delta _p v = \sigma \, v^{p-1}\), \(v\ge 0\) in \(\Omega \), and \(v=1\) on \(\partial \Omega \), which is interesting in its own right.

     

Existence and Uniqueness of Renormalized Solutions to Nonlinear Parabolic Equations with Lower Order Term and Diffuse Measure Data

Here we give an existence and uniqueness result of a renormalized solution for a class of nonlinear parabolic equations \(\displaystyle {\partial b(u) \over \partial t} - \mathrm{div}(a(x,t,\nabla u))+\mathrm{div}(\Phi (x,t, u))=\mu \), where the right side is a measure data, b is a strictly increasing \(C^1\)-function, \(- \mathrm{div}(a(x,t,\nabla u))\) is a Leray–Lions type operator with growth \(|\nabla u|^{p-1}\) in \(\nabla u\) and \(\Phi (x,t, u)\) is a nonlinear lower order term.     

Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

We study the second-order nonlinear differential equation \(u'' + a(t) g(u) = 0\), where \(g\) is a continuously differentiable function of constant sign defined on an open interval \(I\subseteq {\mathbb R}\) and \(a(t)\) is a sign-changing weight function. We look for solutions \(u(t)\) of the differential equation such that \(u(t)\in I,\) satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for \(I = {\mathbb R}^+_0\) and \(g(u) \sim - u^{-\sigma },\) as well as the case of exponential nonlinearities, for \(I = {\mathbb R}\) and \(g(u) \sim \exp (u)\). The proofs are obtained by passing to an equivalent equation of the form \(x'' = f(x)(x')^2 + a(t)\).     

Anti-periodic mild solutions to semilinear fractional differential equations

In this paper, by using the \(\alpha \)-resolvent family theory, Banach contraction mapping principle and Schauder’s fixed point theorem, we investigate the existence of anti-periodic mild solutions to the semilinear fractional differential equations \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t)),\ t\in R,1 \le \alpha \le 2 \) and \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t),u'(t)),\ t\in R,1 < \alpha < 2\), where \(A : D(A)\subset X \rightarrow X\) is the infinitesimal generator of an \(\alpha \)-resolvent family defined on a Banach space \(X\) and \(f\) is a suitable function. Furthermore, an example is given to illustrate our results.     

On Locally Gabriel Geometric Graphs

Let \(P\) be a set of \(n\) points in the plane. A geometric graph \(G\) on \(P\) is said to be locally Gabriel if for every edge \((u,v)\) in \(G\), the Euclidean disk with the segment joining \(u\) and \(v\) as diameter does not contain any points of \(P\) that are neighbors of \(u\) or \(v\) in \(G\). A locally Gabriel graph(LGG) is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique LGG on a given point set since no edge in a LGG is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. The unit distance graph(UDG), introduced by Erdos, is also a LGG. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of LGG: (i) For any \(n\), there exists LGG with \(\Omega (n^{5/4})\) edges. This improves upon the previous best bound of \(\Omega (n^{1+\frac{1}{\log \log n}})\). (ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of LGG. (iii) For any LGG on any \(n\) point set, there exists an independent set of size \(\Omega (\sqrt{n}\log n)\).     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.     

An $$H^1$$-Galerkin mixed finite element method for time fractional reaction–diffusion equation

In this article, an \(H^1\)-Galerkin mixed finite element (MFE) method for solving time fractional reaction–diffusion equation is presented. The optimal time convergence order \(O(\varDelta t^{2-\alpha })\) and the optimal spatial rate of convergence in \(H^1\) and \(L^2\)-norms for variable \(u\) and its gradient \(\sigma \) are derived. Moreover, some numerical results are shown to support our theoretical analysis.     

Estimates for large solutions of a quasilinear elliptic equation

Let \(\Omega \) be a bounded smooth domain of \(R^{n}\). We study the asymptotic behaviour of the solutions to the equation \(\triangle u-|Du|^{q}=f(u)\) in \(\Omega , 1<q<2,\) which satisfy the boundary condition \(u(x)\rightarrow \infty \) as \(x\rightarrow \partial \Omega \). These solutions are called large or blowup solutions. Near the boundary we give lower and upper bounds for the ratio \(\psi (u)/\delta \), where \(\psi (u) = \int _{u}^{\infty }1/\sqrt{2F}dt\), \(F'=f\), \(\delta =dist(x,\partial \Omega )\) or for the ratio \(u/\delta ^{(2-q)/(1-q)}\). When in particular the ratio \(f/F^{q/2}\)is regular at infinity, we find again known results (Bandle and Giarrusso, in Adv Diff Equ 1, 133–150, 1996; Giarrusso, in Comptes Rendus de l’Acad Sci 331, 777–782 2000).     

A Heroin Epidemic Model: Very General Non Linear Incidence,Treat-Age,and Global Stability

The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems

$$ - \mbox{div} \bigl(\epsilon^{2}\phi\bigl(\epsilon|\nabla u|\bigr)\nabla u \bigr) + V(x)\phi\bigl(\vert u\vert\bigr)u = f(u)\quad\mbox{in } \mathbb{R}^{N}, $$

where \(\epsilon\) is a positive parameter, \(N\geq2\), \(V\), \(f\) are continuous functions satisfying some technical conditions and \(\phi\) is a \(C^{1}\)-function.

     

The effect of the domain topology on the number of solutions of fractional Laplace problems

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^n\) is an open bounded set with continuous boundary, \(n>2s\) with \(s\in (0,1),(-\Delta )^{s}\) is the fractional Laplacian operator, \(\mu \) is a positive real parameter, \(q\in [2, 2^*_s)\) and \(2^*_s=2n/(n-2s)\) is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of \(\Omega \). Precisely, we show that the problem has at least \(cat_{\Omega }(\Omega )\) nontrivial solutions, provided that \(q=2\) and \(n\geqslant 4s\) or \(q\in (2, 2^*_s)\) and \(n>2s(q+2)/q\), extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

     

The solution gap of the Brezis–Nirenberg problem on the hyperbolic space

We consider the positive solutions of the nonlinear eigenvalue problem \(-\Delta _{\mathbb {H}^n} u = \lambda u + u^p, \) with \(p=\frac{n+2}{n-2}\) and \(u \in H_0^1(\Omega ),\) where \(\Omega \) is a geodesic ball of radius \(\theta _1\) on \(\mathbb {H}^n.\) For radial solutions, this equation can be written as an ordinary differential equation having n as a parameter. In this setting, the problem can be extended to consider real values of n. We show that if \(2<n<4\) this problem has a unique positive solution if and only if \(\lambda \in \left( n(n-2)/4 +L^*\,,\, \lambda _1\right) .\) Here \(L^*\) is the first positive value of \(L = -\ell (\ell +1)\) for which a suitably defined associated Legendre function \(P_{\ell }^{-\alpha }(\cosh \theta ) >0\) if \(0 < \theta <\theta _1\) and \(P_{\ell }^{-\alpha }(\cosh \theta _1)=0,\) with \(\alpha = (2-n)/2\).     

A Polynomial-Type Jost Solution and Spectral Properties of a Self-Adjoint Quantum-Difference Operator

In this paper, we find a polynomial-type Jost solution of a self-adjoint \(q\)-difference equation of second order. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. We prove that the self-adjoint operator \(L\) generated by the \(q\)-difference expression of second order has essential spectrum filling the segment \([-2\sqrt{q},2\sqrt{q}]\), \(q>1\). Finally, we examine the properties of the eigenvalues of \(L\).     

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.     

Flatness implies smoothness for solutions of the porous medium equation

One of the major problems in the theory of the porous medium equation \(\partial _t\rho =\Delta _x\rho ^m,\,m > 1\), is the regularity of the solutions \(\rho (t,x)\ge 0\) and the free boundaries \(\Gamma =\partial \{(t,x): \rho >0\}\). Here we assume flatness of the solution and derive \(C^\infty \) regularity of the interface after a small time, as well as \(C^\infty \) regularity of the solution in the positivity set and up to the free boundary for some time interval. The proof starts from Caffarelli’s blueprint of an improvement of flatness by rescaling, and combines it with the Carleson measure approach applied to the degenerate subelliptic equation satisfied by the pressure of the porous medium equation in transformed coordinates. The improvement of flatness finally hinges on Gaussian estimates for the subelliptic problem. We use these facts to prove the following eventual regularity result: solutions defined in the whole space with compactly supported initial data are smooth after a finite time \(T_r\) that depends on \(\rho _0\). More precisely, we prove that for \(t \ge T_r\) the pressure \(\rho ^{m-1}\) is \(C^\infty \) in the positivity set and up to the free boundary, which is a \(C^\infty \) hypersurface. Moreover, \(T_r\) can be estimated in terms of only the initial mass and the initial support radius. This regularity result eliminates the assumption of non-degeneracy on the initial data that has been carried on for decades in the literature. Let us recall that regularization for small times is false, and that as \(t\rightarrow \infty \) the solution increasingly resembles a Barenblatt function and the support looks like a ball.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).