Interaction between fast diffusion and geometry of domain

In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type $$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$ under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0 0\) and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.

Several new exact solutions for a fast diffusion equation by the differential constraints of the linear determining equations

We prove existence and uniqueness of the branch of the so-called anomalous eternal solutions in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term ∂tu=Δum+|x|σup, posed in RN with N ⩾ 3, where 1,$?> 0<m<mc=N−2N,p>1, and the critical value for the weight σ=2(p−1)1−m. The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a change of sign of both self-similar exponents at m = m s = (N − 2)/(N + 2), leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a perfect equilibrium in the competition between the fast diffusion and the reaction effects.

We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed. Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined.

The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries

In this paper we study the existence of infinitely many solutions for the second-order differential equation (φp(u′(t)))′−a(t)φq(u(t))+b(t)φr(u(t))=0, t∈(0,T). coupled with Dirichlet’s boundary conditions u (0) = u(T) = 0. Here φ(t) = |t|p-2t < q < p, 1 < r < p and a,b are positive continuous functions on [0, T]. An existence theorem is proved using the variational method and the generalized Clark’s theorem due to Liu and Wang.

The paper discusses the asymptotic behaviour of all solutions of the differential equation , t ∈ I = [t0, ∞), with a positive continuous function a, continuous functions bi, f, and n continuously differentiable unbounded lags. We establish conditions under which any solution y of this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.

We consider the problem of the existence and finite dimensionality of attractors for some classes of two-dimensional turbulent boundarydriven flows that naturally appear in lubrication theory. The flows admit mixed, non-standard boundary conditions and time-dependent driving forces. We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions. Introduction This work gives a survey of the results obtained in a series of papers by Boukrouche & Łukaszewicz (2004, 2005a,b, 2007) and Boukrouche, Łukaszewicz, & Real (2006) in which we consider the problem of the existence and finite dimensionality of attractors for some classes of twodimensional turbulent boundary-driven flows (Problems I–IV below). The flows admit mixed, non-standard boundary conditions and also time-dependent driving forces (Problems III and IV). We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions. This research is motivated by problems from lubrication theory. Our results generalize some earlier ones devoted to the existence of attractors and estimates of their dimensions for a variety of Navier–Stokes flows. We would like to mention a few results that are particularly relevant to the problems we consider. Most earlier results on shear flows treated the autonomous Navier–Stokes equations. In Doering & Wang (1998), the domain of the flow is an elongated rectangle ω = (0, L ) × (0, h ), L ≫ h .

In this paper, we investigate the existence and regularity of positive solutions for certain singular problems that involve an anisotropic ( p , q ) -Laplacian-type operator and a singular term with a variable exponent, under zero Dirichlet boundary conditions on ∂ Ω . The main equation we analyze is − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | p i − 2 ∂ i u ( x ) ) − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | q i − 2 ∂ i u ( x ) ) = f ( x ) u ( x ) γ ( x ) in Ω , where Ω is a bounded, regular domain in R N , f is a positive function belonging to a specific Lebesgue space, and γ ( x ) is a positive continuous function on Ω ¯ . In our study, we do not make comparisons between p i and q i , and as a result, we show that the solution belongs to either W 0 1 , p → ( Ω ) ∩ W 0 1 , q → ( Ω ) or W l o c 1 , p → ( Ω ) ∩ W l o c 1 , q → ( Ω ) depending on the summability of f ( x ) and the values of γ ( x ) . The results are achieved using approximation techniques that include truncation, comparison, and variational methods.

Scalar hyperbolic balance laws in several space dimensions play a central role in this thesis. First, we deal with a new class of mixed parabolic–hyperbolic systems on all Rn: we obtain the basic well–posedness theorems, devise an ad hoc numerical algorithm, prove its convergence and investigate the qualitative properties of the solutions. The extension of these results to bounded domains requires a deep understanding of the initial boundary value problem (IBVP) for hyperbolic balance laws. The last part of the thesis provides rigorous estimates on the solution to this IBVP, under precise regularity assumptions. In Chapter 1 we introduce a predator–prey model. A non local and non linear balance law is coupled with a parabolic equation: the former describes the evolution of the predator density, the latter that of prey. The two equations are coupled both through the convective part of the balance law and the source terms. The drift term is a non local function of the prey density. This allows the movement of predators to be directed towards the regions where the concentration of prey is higher. We prove the well–posedness of the system, hence the existence and uniqueness of solution, the continuous dependence on the initial data and various stability estimates. In Chapter 2 we devise an algorithm to compute approximate solutions to the mixed system introduced above. The balance law is solved numerically by a Lax–Friedrichs type method via dimensional splitting, while the parabolic equation is approximated through explicit finite–differences. Both source terms are integrated by means of a second order Runge–Kutta scheme. The key result in Chapter 2 is the convergence of this algorithm. The proof relies on a careful tuning between the parabolic and the hyperbolic methods and exploits the non local nature of the convective part in the balance law. This algorithm has been implemented in a series of Python scripts. Using them, we obtain information about the possible order of convergence and we investigate the qualitative properties of the solutions. Moreover, we observe the formation of a striking pattern: while prey diffuse, predators accumulate on the vertices of a regular lattice. The analytic study of the system above is on all Rn. However, both possible biological applications and numerical integrations suggest that the boundary plays a relevant role. With the aim of studying the mixed hyperbolic–parabolic system in a bounded domain, we noticed that for balance laws known results lack some of the estimates necessary to deal with the coupling. In Chapter 3 we then focus on the IBVP for a general balance law in a bounded domain. We prove the well–posedness of this problem, first with homogeneous boundary condition, exploiting the vanishing viscosity technique and the doubling of variables method, then for the non homogeneous case, mainly thanks to elliptic techniques. We pay particular attention to the regularity assumptions and provide rigorous estimates on the solution.

A rigorous methodology for the analysis of initial boundary value problems on the half-line, 0 < x < ∞, t > 0, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the solution q(x, t) of the sine-Gordon equation can be obtained in terms of the solution of a 2 × 2 matrix Riemann–Hilbert problem. This problem is formulated in the complex k-plane and is uniquely defined in terms of the so-called spectral functions a(k), b(k), and B(k)/A(k). The functions a(k) and b(k) can be constructed in terms of the given initial conditions q(x, 0) and qt(x, 0) via the solution of a system of two linear ODEs, while for arbitrary boundary conditions the functions A(k) and B(k) can be constructed in terms of the given boundary condition via the solution of a system of four nonlinear ODEs. In this paper, we analyse two particular boundary conditions: the case of constant Dirichlet data, q(0, t) = χ, as well as the case when qx(0, t), sin (q(0, t)/2), and cos(q(0, t)/2) are linearly related by two constants χ1 and χ2. We show that for these particular cases, the system of the above nonlinear ODEs can be avoided, and B(k)/A(k) can be computed explicitly in terms of { a(k), b(k), χ} and {a(k), b(k), χ1, χ2}, respectively. Thus, these ‘linearizable’ initial boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line.

This paper is concerned with the existence of solutions to the class of nonlocal boundary value problems of the type -M\left(\int_{\Omega}|\nabla u|^2\right)\Delta u = f(x,u), \text { in } \Omega, \ u=0, \text{ on } \partial \Omega, where \Omega is a smooth bounded domain of \mathbb{R}^N and M is a positive continuous function. By assuming that f(x, u) is a Carathéodory function which growths at most |u|^{\frac{N}{N-2}} , N \geq 3 , and under a suitable growth condition on M , one proves an existence result by applying the Leray–Schauder fixed point theorem.

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. &#x0D; Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems.&#x0D; Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. &#x0D; Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations.&#x0D; Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

We study the Cauchy problem and the initial boundary value problem (IBVP) for nonlinear parabolic equations in $\\mathcal{C}_b([0,T);L^p)$ and $L^q(0,T;L^p)$.\nWe give a unified method to construct local mild solutions of the Cauchy problem or IBVP for a class of nonlinear parabolic equations in $\\mathcal{C}_b([0,T);L^p)$ or $L^q(0,T;L^p)$ by introducing admissible triplet, generalized admissible triplet and establishing time space estimates for the solutions to the linear parabolic equations.\nMoreover, using our method, we also obtain the existence of global small solutions to the nonlinear parabolic equations.