Existence and Regularity of the Density for Solutions to Semilinear Dissipative Parabolic SPDEs

We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on $L^2(\mathcal{O})$ (evaluated at fixed points in time and space), where $\mathcal{O}$ is an open bounded domain in ? ^d with smooth boundary. The equation is driven by an additive Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be (maximal) monotone, continuously differentiable, and growing not faster than a polynomial. The proof uses tools of the Malliavin calculus combined with methods coming from the theory of maximal monotone operators.     

Maximal saddle solution of a nonlinear elliptic equation involving the p-Laplacian

A saddle solution is called maximal saddle solution if its absolute value is not smaller than those absolute values of any solutions that vanish on the Simons cone $\mathcal{C}=\{s=t\}$ and have the same sign as s???t. We prove the existence of a maximal saddle solution of the nonlinear elliptic equation involving the p-Laplacian, by using the method of monotone iteration, $$ \Delta_{p}u=f(u) \quad \text{in} \quad R^{2m}, $$ where 2m?≥?p?>?2.     

Regularization and iterative methods for monotone inverse variational inequalities

We consider the monotone inverse variational inequality: find $x\in H$ such that $$\begin{aligned} f(x)\in \Omega , \quad \left\langle \tilde{f}-f(x),x\right\rangle \ge 0, \quad \forall \tilde{f}\in \Omega , \end{aligned}$$ where $\Omega $ is a nonempty closed convex subset of a real Hilbert space $H$ and $f:H\rightarrow H$ is a monotone mapping. A general regularization method for monotone inverse variational inequalities is shown, where the regularizer is a Lipschitz continuous and strongly monotone mapping. Moreover, we also introduce an iterative method as discretization of the regularization method. We prove that both regularized solution and an iterative method converge strongly to a solution of the inverse variational inequality.     

Translation invariant diffusions in the space of tempered distributions

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σ _ij , b _i and initial condition y in the space of tempered distributions) that may be viewed as a generalisation of Ito’s original equations with smooth coefficients. The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σ _ij $\tilde y$ , b _i $\tilde y$ are assumed to be locally Lipshitz.Here denotes convolution and $\tilde y$ is the distribution which on functions, is realised by the formula $\tilde y\left( r \right): = y\left( { - r} \right)$ . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.     

A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let ?? >?0 and let A be an ??-inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H. Let F be a maximal monotone operator on H such that the domain of F is included in C. Let 0?< k?<?1 and let g be a k-contraction of H into itself. Let V be a ${\overline{\gamma}}$ -strongly monotone and L-Lipschitzian continuous operator with ${\overline{\gamma} >0 }$ and L >?0. Take ${\mu, \gamma \in \mathbb R}$ as follows: $${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$ In this paper, under the assumption ${(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}$ , we prove a strong convergence theorem for finding a point ${z_0\in (A+B)^{-1}0\cap F^{-1}0}$ which is a unique solution of the hierarchical variational inequality $${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$ Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.     

On the Behavior of Gegenbauer Polynomials in the Complex Plane

It is well-known that the squared modulus of every function f from the Laguerre–Polya class ${\mathcal{L}-\mathcal{P}}$ of entire functions obeys a MacLaurin series representation $$|f(x+i y)|^2=\sum_{k=0}^{\infty} L_k(f;x)\,y^{2k}, \quad x,y\in\mathbb{R}$$ , which reduces to a finite sum when f is a polynomial having only real zeros. The coefficients {L _k } are representable as non-linear differential operators acting on f, and by a classical result of Jensen L _k (f;x)?≥ 0 for ${f\in \mathcal{L}-\mathcal{P}}$ and ${x\in \mathbb{R}}$ . Here, we prove a conjecture formulated by the first-named author in 2005, which states that for ${f=P_n^{(\lambda)} }$ , the n-th Gegenbauer polynomial, the functions ${\{L_k(f;x)\}_{k=1}^{n}}$ are monotone decreasing on the negative semi-axis and monotone increasing on the positive semi-axis. This result pertains to certain polynomial inequalities in the spirit of the celebrated refinement of Markov’s inequality, found by R. J. Duffin and A. C. Schaeffer in 1941.     

Strong convergence of a splitting projection method for the sum of maximal monotone operators

In this paper, a projective-splitting method is proposed for finding a zero of the sum of $n$ maximal monotone operators over a real Hilbert space $\mathcal{H }$ . Without the condition that either $\mathcal{H }$ is finite dimensional or the sum of $n$ operators is maximal monotone, we prove that the sequence generated by the proposed method is strongly convergent to an extended solution for the problem, which is closest to the initial point. The main results presented in this paper generalize and improve some recent results in this topic.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

The Algebraicity of Ill-Distributed Sets

We show that every set \({S \subseteq [N]^d}\) occupying \({\ll p^{\kappa}}\) residue classes for some real number \({0 \leq \kappa < d}\) and every prime p, must essentially lie in the solution set of a polynomial equation of degree \({\ll ({\rm log} N)^C}\) , for some constant C depending only on \({\kappa}\) and d. This provides the first structural result for arbitrary \({\kappa < d}\) and S.     

On the supercritical KdV equation with time-oscillating nonlinearity

For the initial value problem (IVP) associated to the generalized Korteweg–de Vries (gKdV) equation with supercritical nonlinearity, $$u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5,$$ numerical evidence [3] shows that, there are initial data ${\phi\in H^1(\mathbb{R})}$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [1, 18], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation $$u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0,$$ where g is a periodic function and ${k\geq 5}$ is an integer. We prove that, for given initial data ${\phi \in H^1(\mathbb{R})}$ , as ${|\omega|\to \infty}$ , the solution ${u_{\omega} }$ converges to the solution U of the initial value problem associated to $$U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0,$$ with the same initial data, where m(g) is the average of the periodic function g. Moreover, if the solution U is global and satisfies ${\|U\|_{L_x^{5}L_t^{10}}<\infty}$ , then we prove that the solution ${u_{\omega} }$ is also global provided ${|\omega|}$ is sufficiently large.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions

In this paper, we are concerned with nonlocal problem for fractional evolution equations with mixed monotone nonlocal term of the form $$\left\{\begin{array}{ll}^CD^{q}_tu(t) + Au(t) = f(t, u(t), u(t)),\quad t \in J = [0, a],\\u(0) = g(u, u),\end{array}\right.$$ where E is an infinite-dimensional Banach space, \({^CD^{q}_t}\) is the Caputo fractional derivative of order \({q\in (0, 1)}\) , A : D(A) ? E → E is a closed linear operator and ?A generates a uniformly bounded C ₀-semigroup T(t) (t ≥  0) in E, \({f \in C(J\times E \times E, E)}\) , and g is appropriate continuous function so that it constitutes a nonlocal condition. Under a new concept of coupled lower and upper mild L-quasi-solutions, we construct a new monotone iterative method for nonlocal problem of fractional evolution equations with mixed monotone nonlocal term and obtain the existence of coupled extremal mild L-quasi-solutions and the mild solution between them. The results obtained generalize the recent conclusions on this topic. Finally, we present two applications to illustrate the feasibility of our abstract results.     

The effective conductivity equation for a highly heterogeneous periodic medium

We consider the homogenization of a conductivity equation for a medium made up of a set ${F_\varepsilon}$ ( ${\varepsilon}$ being the size of the period of the medium) of highly conductive vertical fibers surrounded by another material (the matrix) assumed to be a poor conductor. The conductivity coefficients in the fibers behave as ${\frac{1}{\varepsilon^2}}$ while whose of the matrix behave as ${\varepsilon^2}$ . We show that the homogenized problem consists of an equality of the kind u(x) = m(x) f (x) where u denotes the macroscopic temperature, f the source term and m(x) a coefficient given by solving some cell equation.     

Global existence and blowup of solutions for a class of nonlinear higher-order wave equations

In this paper, we consider a class of nonlinear higher-order wave equation with nonlinear damping $$u_{tt}+(-\Delta)^mu+a|u_t|^{p-2}u_t=b|u|^{q-2}u$$ in a bounded domain ${\Omega\subset\mathbb{R}^N}$ (N????1 is a natural number). We show that the solution is global in time under some conditions without the relation between p and q and we also show that the local solution blows up in finite time if q?>?p with some assumptions on initial energy. The decay estimate of the energy function for the global solution and the lifespan for the blow-up solution are given. This extend the recent results of Ye (J Ineq Appl, 2010).     

Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications to Initial Value Problems

This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$ in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$ where ${t \in [0, T]}$ is the time variable, x runs in a bounded and simply connected domain in ${\mathbb{R}^{4}, \lambda}$ is a real number, and ${\mathcal{D}}$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}_{\lambda}}$ , i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\lambda}u = 0}$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators ${\mathcal{L}}$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u ₀ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.     

Preserving maximal monotonicity with applications in sum and composition rules

In this paper, we study maximal monotonicity preserving mappings on the Banach space X × X ^*. Indeed, for a maximal monotone set ${M \subset X\times X^*}$ and for a multifunction ${T: X \times X^* \multimap Y \times Y^*}$ , under some sufficient conditions on M and T we show that T(M) is maximal monotone. As two consequences of this result we get sum and composition rules for maximal monotone operators.     

The Properties of Solutions for a Generalized b-Family Equation with Peakons

This paper deals with the Cauchy problem for a shallow water equation with high-order nonlinearities, y _t +u ^m+1 y _x +bu ^m u _x y=0, where b is a constant, $m\in \mathbb{N}$ , and we have the notation $y:= (1-\partial_{x}^{2}) u$ , which includes the famous Camassa–Holm equation, the Degasperis–Procesi equation, and the Novikov equation as special cases. The local well-posedness of strong solutions for the equation in each of the Sobolev spaces $H^{s}(\mathbb{R})$ with $s>\frac{3}{2}$ is obtained, and persistence properties of the strong solutions are studied. Furthermore, although the $H^{1}(\mathbb{R})$ -norm of the solution to the nonlinear model does not remain constant, the existence of its weak solutions in each of the low order Sobolev spaces $H^{s}(\mathbb{R})$ with $1<s<\frac{3}{2}$ is established, under the assumption $u_{0}(x)\in H^{s}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$ . Finally, the global weak solution and peakon solution for the equation are also given.     

On minimum norm solutions

This note investigates the problem $$\min x_p^p /p,s.t.Ax \geqslant b,$$ where 1<p<∞. It is proved that the dual of this problem has the form $$\max b^T y - A^T y_q^q /q,s.t.y \geqslant 0,$$ whereq=p/(p?1). The main contribution is an explicit rule for retrieving a primal solution from a dual one. If an inequality is replaced by an equality, then the corresponding dual variable is not restricted to stay nonnegative. A similar modification exists for interval constraints. Partially regularized problems are also discussed. Finally, we extend an observation of Luenberger, showing that the dual of $$\min x_p ,s.t.Ax \geqslant b,$$ is $$\max b^T y,s.t.y \geqslant 0,A^T y_q \leqslant 1,$$ and sharpening the relation between a primal solution and a dual solution.     

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

We study the Cauchy–Dirichlet problem for the elliptic–parabolic equation $$b(u)_t + {\rm div} F(u) - \Delta u = f$$ in a bounded domain. We do not assume the structure condition $$b(z) = b(\hat z) \Rightarrow F(z) = F(\hat z).$$ Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold (Proc R Soc Edinb 133A(3):477–496, 2003) where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux F, we justify the uniqueness of u (the uniqueness of b(u) is well-known) and prove the continuous dependence in L ¹ for the case of strongly convergent finite energy data. We also prove convergence of the ${\varepsilon}$ -discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux F, we show that the problem admits a maximal and a minimal solution.     

Existence of nontrivial solutions of linear functional equations

In this paper we study the functional equation $$\sum_{i=1}^n a_i f(b_i x+c_i h)=0 \quad (x, h \in \mathbb{C})$$ where a _i , b _i , c _i are fixed complex numbers and \({f \colon \mathbb{C} \to \mathbb{C}}\) is the unknown function. We show, that if there is i such that \({b_i / c_i \neq b_j /c_j}\) holds for any \({1 \leq j \leq n,\ j \neq i}\) , the functional equation has a nonconstant solution if and only if there are field automorphisms \({\phi_1, \ldots, \phi_k}\) of \({\mathbb{C}}\) such that \({\phi_1 \cdots \phi_k}\) is a solution of the equation.