Dirichlet Problem for Stationary Anisotropic Higher-Order Partial Integrodifferential Equations with Variable Exponents of Nonlinearity

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

We consider parabolic variational inequalities in Sobolev spaces and generalized Lebesgue spaces. Sufficient conditions of existence and uniqueness of solution of these inequalities have been established. We have also proved some comparison theorems.

The initial-boundary value problems for parabolic equations with variable exponents of nonlinearity and time depended delay are considered. Existence and uniqueness of solutions of these problems are proved.

In Sobolev spaces with variable exponent, we consider the problem for a semilinear hyperbolic variational inequality of the third order. We establish conditions for the existence of a solution u of this problem such that u ∈ L ∞((0, T); V 1,0(Ω)), u t ∈ L ∞((0, T); V 1,0(Ω)) ∩ L p(x)(Q T ), and u tt ∈ L ∞((0, T); L 2(Ω)), where V 1,0(Ω) ⊂ H 1(Ω).

On the well-posedness of a Fourier problem for nonlinear parabolic equations of higher order with variable exponents of nonlinearity

We consider the following nonlinear parabolic equation: ut-div(|∇u|p(x)-2∇u)=f(x,t), where f:Ω×(0,T)→R and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.

Рассматривается вопрос существования решений задачи Дирихле для нелинейных эллиптических уравнений второго порядка в неограниченных областях. Ограничения на структуру квазилинейных уравнений формулируются в терминах специального класса выпуклых функций - обобщенных $N$-функций. А именно, нелинейности определяются функциями Музилака-Орлича такими, что дополнительные к ним функции подчиняются $\Delta_2$-условию. Соответствующее пространство Музилака-Орлича-Соболева не обязано быть рефлексивным. Именно этот факт является существенной проблемой, поскольку теорема для псевдомонотонных операторов здесь не применима. Для рассматриваемого класса уравнений доказательство теоремы существования проводится на основе абстрактной теоремы для дополнительных систем. Важным инструментом, который позволил обобщить имеющиеся результаты существования решений рассматриваемых уравнений для ограниченных областей на неограниченные области, является теорема вложения пространств Музилака-Орлича-Соболева. Таким образом, в работе найдены условия на структуру квазилинейных уравнений в терминах функций Музилака-Орлича, достаточные для разрешимости задачи Дирихле в неограниченных областях. Кроме того, приведены примеры уравнений, показывающие, что класс нелинейностей, рассматриваемый в работе, шире, чем нестепенные и степенные нелинейности.

The paper is focused on solvability of an inverse boundary problem with an unknown coefficient which depends on time for a second-order parabolic equation with non-classical boundary conditions. The idea of the problem is that together with the solution it is required to determine the unknown coefficient. The problem is considered in the rectangular area. The paper introduces a classical solution of the set problem. At first, an auxiliary inverse boundary problem is examined and the equivalence (in some sense) of the original problem is proved. First, we apply a method of variable separation to analyze the auxiliary inverse boundary problem. Then, we examine a spectral problem for an ordinary second-order differential equation with integral conditions. Having used a formal scheme of the method of variable separation, the solution of a direct boundary problem (in case of specified unknown function) resolves itself into solution of Cauchy problem. After that the solution is limited to the solution of a countable system of integro-differential equations in Fourier coefficients. In its turn, the last system regarding unknown Fourier coefficients is recorded in the form of an integro-differential equation in the desired solution. Using relevant additional conditions of the auxiliary inverse boundary problem, we obtain a system of two nonlinear integral equations for defining unknown functions. Thus, the solution of the auxiliary inverse boundary problem comes down to the system of two nonlinear integro-differential equations in unknown functions. The specific Banach space is designed. Then, in the sphere made of the Banach space we with the help of contracted mapping prove the solvability of the nonlinear integro-differential equations set, which is a unique solution of the additional inverse boundary problem. Using the equivalence of problems, it is concluded about existence and uniqueness of a classical solution of the original problem.

In this paper, a first-order equation with state-dependent delay and with a nonlinear right-hand side is considered. Conditions of existence and uniqueness of the solution of initial value problem aresupposed to be executed. The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. Local dynamics depends on real parameters which are coefficients of equation right-hand side decomposition in a Taylor series. The parameter which is a coefficient at the linear part of this decomposition has two critical values which determine a stability domain of zero equilibrium. We introduce a small positive parameter and use the asymtotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov – Hopf bifurcation occurs near the second of them (if the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered as an example. The bifurcation parameter of this equation has one critical value. A simple sufficient condition of Andronov – Hopf bifurcation occurence in the considered equation near a critical value is obtained as a result of applying the method of normal forms.

Existence and uniqueness of relative incidence estimates in case-series analysis

We study conditions for existence, uniqueness, and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al. (Funding valuation adjustment: a consistent framework including CVA, DVA, collateral, netting rules and re-hypothecation, 2011, [11]). These equations take the form of semi-linear PDEs and Forward–Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage-free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk-free rate. Indeed, our final semi-linear PDE or FBSDEs and their classical solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk-free rate with a real market rate, since it acts as an instrumental variable. The equations’ derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including Brigo and Pallavicini (J. Financ. Eng. 1(1):1–60 (2014), [3]), Pallavicini and Brigo (Interest-rate modelling in collateralized markets: multiple curves, credit-liquidity effects, CCPs, 2011, [10]), Pallavicini et al. (Funding valuation adjustment: a consistent framework including cva, dva, collateral, netting rules and re-hypothecation, 2011, [11]), Pallavicini et al. (Funding, collateral and hedging: uncovering the mechanics and the subtleties of funding valuation adjustments, 2012, [12]), and Brigo et al. (Nonlinear valuation under collateral, credit risk and funding costs: a numerical case study extending Black–Scholes, [5]).

For a class of second-order anisotropic elliptic equations with variable nonlinearity indices and summable right-hand sides, we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable exponents.

The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random conditions is a classical problem of mathematical physics. In this paper we consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space Subφ(Ω). The conditions of existence with probability one generalized solution of the problem are investigated.

In the preceding chapter, we developed rather completely the regularity properties of weak and strong solutions of certain first order variational problems and second order differential equations involving only one unknown function. Complete results concerning the solutions of the corresponding problems involving several unknown functions were obtained only in the case where v = 2; these results were obtained by the writer before the war and were described in Chapter 1. In the preceding chapter some first differentiability results were obtained for certain systems of equations but these results did not imply the continuity of the first derivatives. In 1952, the writer presented a paper (Morrey [10]) at the Arden House Conference on Partial Differential Equations, in which it was proved that any vector solution of class C1 of a regular variational problem of class C n μ , n ≥ 2, was also of class C n μ . These results still leave a gap in the theory for systems, which can only be filled by an extension of the De Giorgi-Nash results, developed in § 5–3, to systems or some entirely new device. But, in proving the C n gm differentiability results for systems of equations, the writer was forced to use the very important formulas of F. John ([1], [3]) for fundamental solutions and to use methods which are appropriate for the discussion of elliptic systems of higher order. Hence the inclusion of this chapter in this book.KeywordsWeak SolutionElliptic SystemGeneral BoundaryPrincipal PartRegularity TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Higher order partial differential equations with functional arguments including hyperbolic equations and beam equations are studied. Sufficient conditions are derived for every solution of certain boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi‐dimensional oscillation problem to a one‐dimensional problem for higher order functional differential inequalities.