Unbounded Initial Data Research Articles

Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data

Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data

Estimates for the Heat Flow in Optimal Spaces of Unbounded Initial Data in mathbb {R}^{{d}} and Applications to the Ornstein–Uhlenbeck Semigroup

In this paper, we improve some estimates from [7] for solutions of the heat equation with initial conditions that are large ‘at infinity’ and employ them to obtain suitable estimates on the solutions of the Ornstein–Uhlenbeck semigroup [solutions of u_t-Delta u=alpha x cdot nabla u] for the same class of unbounded data.

The heat flow in an optimal Fréchet space of unbounded initial data in [formula omitted

In this paper we show that solutions of the heat equation that are given in terms of the heat kernel define semigroups on the family of Fréchet spaces L0p(Rd), the intersection (over all ε>0) of the spaces Lεp(Rd) of functions such that ∫Rde−ε|x|2|f(x)|pdx<∞. These spaces consist of functions that are ‘large at infinity’, and L01(Rd) is the maximal space in which one can use the heat kernel to obtain globally-defined solutions of the heat equation. We prove suitable estimates from L0p(Rd) into L0q(Rd), q≥p, for these semigroups.We then consider the heat semigroup posed in spaces that are dual to these spaces of functions, namely the spaces L−εp(Rd) of very-rapidly decreasing functions such that ∫Rdeε|x|2|f(x)|pdx<∞. We show that (Lpεp(Rd))′=L−qεq(Rd) (with 1<p<∞ and (p,q) conjugate), and that the heat flow on Lεp(Rd) is the adjoint of the flow on L−δq(Rd) for an appropriate (time-dependent) choice of δ.

Nonuniqueness of Unbounded Solutions of the Cauchy Problem for Scalar Conservation Laws

This article is aimed at studying the Cauchy problem for a first-order quasi-linear equation with a flow function of power type and unbounded initial data of power or exponential type. It is known that the Cauchy problem in the class of locally bounded functions may have several solutions. We describe all entropy solutions of this problem, which can be represented in a special form. It is shown that after the first discontinuity line (shock wave), these solutions eventually exhibit the same behavior, and their nonuniqueness actually amounts to the choice of the first shock wave.

Multi-dimensional Burgers Equation with Unbounded Initial Data: Well-Posedness and Dispersive Estimates

The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{t\ge0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t)_{t\ge0}$ extends uniquely as a continuous semi-group over $L^p(\mathbb{R}^n)$ whenever $1\le p<\infty$, and $u(t):=S_tu_0$ is actually an entropy solution to the Cauchy problem. When $p\le q\le \infty$ and $t>0$, $S_t$ actually maps $L^p(\mathbb{R}^n)$ into $L^q(\mathbb{R}^n)$. These results are based upon new dispersive estimates. The ingredients are on the one hand Compensated Integrability, and on the other hand a De Giorgi-type iteration.

The Optimal Convergence Rate of Monotone Schemes for Conservation Laws in the Wasserstein Distance

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, $\Lip^+$-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of $\Lip^+$-unbounded initial data is worse than first-order.

Local and global time decay for parabolic equations with super linear first-order terms

We study a class of parabolic equations having first order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.

Porous medium equations on manifolds with critical negative curvature: unbounded initial data

ABSTRACTWe investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan–Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity.

Example of nonexistence of a positive generalized entropy solution of a Cauchy problem with unbounded positive initial data

A Cauchy problem for a first-order quasilinear equation with polynomial flux function and exponential initial data is studied. We prove that this problem has no positive solution in the class of locally bounded functions.

Slow growth of solutions of superfast diffusion equations with unbounded initial data

Slow growth of solutions of superfast diffusion equations with unbounded initial data

Local and global existence of solutions to a quasilinear degenerate chemotaxis system with unbounded initial data

This paper is concerned with local and global existence of solutions to the parabolic‐elliptic chemotaxis system . Marinoschi (J. Math. Anal. Appl. 2013; 402:415–439) established an abstract approach using nonlinear m‐accretive operators to giving existence of local solutions to this system when 0 &lt; D0≤D′(r)≤D∞&lt;∞ and (r1,r2)↦K(r1,r2)r1 is Lipschitz continuous on , provided that the initial data is assumed to be small. The smallness assumption on the initial data was recently removed (J. Math. Anal. Appl. 2014; 419:756–774). However the case of non‐Lipschitz and degenerate diffusion, such as D(r) = rm(m &gt; 1), is left incomplete. This paper presents the local and global solvability of the system with non‐Lipschitz and degenerate diffusion by applying (J. Math. Anal. Appl. 2013; 402:415–439) and (J. Math. Anal. Appl. 2014; 419:756–774) to an approximate system. In particular, the result in the present paper does not require any properties of boundedness, smoothness and radial symmetry of initial data. This makes it difficult to deal with nonlinearity. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Solvability of the Cauchy Problem for a Parabolic Equation with Unbounded Coefficients

We consider the Cauchy problem in the half-space for a parabolic equation with unbounded initial data and lower order coefficients. We establish the solvability of the problem in the weighted space of Holder functions such that the functions, together with their derivatives, admit exponential growth as t → ∞ and, near the plane t = 0, the derivatives grow not faster than a power function. Bibliography: 11 titles.

Solutions of a system of forced Burgers equation

In this article, we obtain explicit solutions of a system of forced Burgers equation subject to some classes of bounded and compactly supported initial data and also subject to certain unbounded initial data. In a series of papers, Rao and Yadav (2010) [1–3] obtained explicit solutions of a nonhomogeneous Burgers equation in one dimension subject to certain classes of bounded and unbounded initial data. Earlier Kloosterziel (1990) [4] represented the solution of an initial value problem for the heat equation, with initial data in L2(Rn,e|x|2/2), as a series of self-similar solutions of the heat equation in Rn. Here we express the solutions of certain classes of Cauchy problems for a system of forced Burgers equation in terms of self-similar solutions of some linear partial differential equations.

A note on entropy solutions for degenerate parabolic equations with L&lt;SUP align="right"&gt;1&lt;/SUP&gt; &amp;cap; L&lt;SUP align="right"&gt;p&lt;/SUP&gt; data

We prove existence and uniqueness results for entropy solutions of degenerate parabolic equations with appropriately growing nonlinearities and unbounded initial data.

Local and global properties of solutions of heat equation with superlinear absorption

We study the limit when $k\to\infty$ of the solutions of $ \partial_tu-\Delta u+f(u)=0$ in $\mathbb R^N\times (0,\infty)$ with initial data $k\delta$, when $f$ is a positive superlinear increasing function. We prove that there exist essentially three types of possible behaviour according to whether $f^{-1}$ and $F^{-1/2}$ belong or not to $L^1(1,\infty)$, where $F(t)=\int_0^t f(s)ds$. We use these results for providing a new and more general construction of the initial trace and some uniqueness and nonuniqueness results for solutions with unbounded initial data.

Uniqueness and comparison properties of the viscosity solution to some singular HJB equations

We study viscosity solutions to a class of HJB equations with singular coefficients near at the boundary: cases with either vanishing, or oscillating, or blowing-up diffusion coefficients are included. Because of proper structural conditions, strong comparison principle holds without assigning spatial boundary data, and unbounded initial data can be handled. The result applies to stochastic models for interest rate, and yields new results concerning Cauchy problems with unbounded coefficients.

Instantaneous shrinking of the support of solutions to certain parabolic equations with unbounded initial data

Instantaneous shrinking of the support of solutions to certain parabolic equations with unbounded initial data

Local existence and uniqueness of weak solutions for nonlinear parabolic equations with superlinear growth and unbounded initial data

Local existence and uniqueness of weak solutions for nonlinear parabolic equations with superlinear growth and unbounded initial data

Benjamin–Ono Equation with Unbounded Data

The initial value problem (IVP) for the Benjamin–Ono equation with unbounded initial data is considered. We show existence and uniqueness of global solutions for sublinear growth data. The method of proof relies in an appropriate splitting, the study of an IVP associated with a Benjamin–Ono type equation with variable coefficients and the persistence of decay for solutions of the Burgers equation.

Direct asymptotic analysis of the second Painlevé equation: three different limits

We present three different asymptotic studies of the second Painlevé equation involving , or unbounded initial data. We show how the direct method, which is in the spirit of Boutroux, can be naturally applied to each of the three cases.