A spatial decay estimate for the heat equation

Zusammenfassung Ein anisotroper, fester Wärmeleiter liege vor, dessen Grenzflächen, mit der Ausnahme einer ebenen OberflächeC, isoliert sind. Es werden Ungleichungen aufgestellt, welche die exponentielle Abnahme des Wärmefeldes im Körper mit der Entfernung vonC ₀ ausdrücken.     

On the spatial decay of solutions of parabolic equations

Zusammenfassung Es wird ein von Edelstein erzieltes Ergebnis über den exponentiellen Temperaturabfall eines Körpers an von Wärmezufuhrquellen entfernten Stellen auf eine erweiterte Klasse von Randwertproblemen verallgemeinert.

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This work supported by the U.S. Department of the Navy Contract NOw-62-0604-c.     

On a decay property of solutions to the Haraux-Weissler equation

We give a sufficient condition that non-radial H¹-solutions to the Haraux-Weissler equation should belong to the weighted Sobolev space , where ρ is the weight function exp(|x|²/4). Our result provides, in some sense, a connection between the solutions obtained by ODE method and those by variational approach in the space .     

On the rate of decay of solutions to linear viscoelastic equation

We investigate the L_p–L_q estimate of solutions to the Cauchy problem of linear viscoelastic equation, especially, the diffusion wave property of linear viscoelastic equation like the Navier–Stokes equation in the compressible fluid case, which was studied by D. Hoff and K. Zumbrum and Tai‐P. Liu and W. Wang. Copyright © 2000 John Wiley & Sons, Ltd.     

A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

We consider the semilinear parabolic equation _ut=Δu+u^p$u_t=\mathrm{\Delta }u+u^p$ on R^N${\mathbb{R}}^N$, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈R^N$x\in {\mathbb{R}}^N$ and t∈R$t\in \mathbb{R}$. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t?0$t?0$, then it necessarily converges to 0, as t→∞$t\to \infty$, uniformly with respect to x∈R^N$x\in {\mathbb{R}}^N$.     

Improved entropy decay estimates for the heat equation

Improved entropy decay estimates for the heat equation are obtained by selecting well-parametrized Gaussians. Either by mass centering or by fixing the second moments or the covariance matrix of the solution, relative entropy toward these Gaussians is shown to decay with better constants than classical estimates.     

On analytic solutions of the heat equation with an operator coefficient

Let A be a bounded linear operator on a Banach space and let g a be vector-valued function that is analytic in a neighborhood of the origin of ℝ. We obtain conditions of the existence of analytic solutions for the Cauchy problem Moreover, we consider a representation of the solution of this problem as a Poisson integral and study the Cauchy problem for the corresponding inhomogeneous equation. Bibliography: 22 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 139–162.     

Uniform decay estimates for solutions of the Yamabe equation

We study positive solutions u of the Yamabe equation c_m Du-s( x) u+k( x) u^\fracm+2m-2=0{c_{m} \Delta u-s\left( x\right) u+k\left( x\right) u^{\frac{m+2}{m-2}}=0}, when k(x) > 0, on manifolds supporting a Sobolev inequality. In particular we get uniform decay estimates at infinity for u which depend on the behaviour at infinity of k, s and the L ^Γ-norm of u, for some G 3 \tfrac2mm-2{\Gamma\geq\tfrac{2m}{m-2}}. The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.     

On the decay rate of solutions of the wave equation in domains with star-shaped boundaries

One studies the large-time decay rate of the weighted energy of solutions of the first mixed problem for the wave equation in domains with smooth boundaries which are star-shaped with respect to the origin. Estimates are established for solutions of the first boundary-value problem for the Helmholtz equation in the upper half-plane. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 390–406, 2007.     

General decay of solutions of a viscoelastic equation

In this paper we consider the following viscoelastic equation:

     

The decay mode solutions for the cylindrical KP equation

The decay mode solutions for the cylindrical Kadomtsev-Petviashvili equation can be obtained by the Bäcklund transformation and Hirota method.     

On positive solutions of the heat equation satisfying a nonlinear boundary condition

We consider a boundary value problem for the heat equation in the exterior of a bounded domain of space variables. On the boundary of the domain, we pose a nonlinear boundary condition. We find sharp nonlinearity exponents for which there exists no global solution.     

Discretization of the solutions of the heat equation

We obtain the exact order of discretization (reconstruction) errors, given linear information on the solutions of the heat equation.     

Asymptotic decay of solutions of the liouville equation under perturbations

We consider the Cauchy problem for a perturbed Liouville equation. An asymptotic solution is constructed with respect to the perturbation parameter by the two-scale expansion method; this construction can be applied over long time intervals. The main result is the definition of a deformation of the leading term of the asymptotic expansion within a slow time scale. Translated frommatematicheskie Zametki, Vol. 68, No. 2, pp. 195–209, August, 2000.