Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝ^N, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration

Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim_{y → 0}k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)u_xx – ∂_y(𝓁(y)u_y) + a(x, y)u_x = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|_AC = 0, where G is a simply connected domain in ℝ² with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫^y₀(k(t)/𝓁(t))^1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L²(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u₁) – f(x, y, u₂| ≤ C|u₁ – u₂|, where C = const > 0.     

Reaction diffusion equations with non-linear boundary conditions,blowup and steady states

In this paper we study the following problem: u_t−Δu=−f(u) in Ω×(0, T)≡Q_T, ∂u ∂n=g(u) on ∂Ω×(0, T)≡S_T, u(x, 0)=u₀(x) in Ω , where Ω⊂ℝ^N is a smooth bounded domain, f and g are smooth functions which are positive when the argument is positive, and u₀(x)>0 satisfies some smooth and compatibility conditions to guarantee the classical solution u(x, t) exists. We first obtain some existence and non-existence results for the corresponding elliptic problems. Then, we establish certain conditions for a finite time blow-up and global boundedness of the solutions of the time-dependent problem. Further, we analyse systems with same kind of boundary conditions and find some blow-up results. In the last section, we study the corresponding elliptic problems in one-dimensional domain. Our main method is the comparison principle and the construction of special forms of upper–lower solutions using related equations. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A Threshold Result for a Non-local Parabolic Equation

In this paper, we consider the Cauchy problem: (ECP) u_t−Δu+p(x)u=u(x,t)∫u²(y,t)/∣x−y∣dy; x∈ℝ³, t>0, u(x, 0)=u₀(x)⩾0 x∈ℝ³, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ³, p(x)⩾ā>0, and lim_{∣x∣→∞}p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u₀(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u₀(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws

This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws u_t + f(u)_x = u_xx on [0, 1], with the boundary condition u(0, t) = u₋(t) → u₋, u(1, t) = u₊(t) → u₊, as t → +∞ and the initial data u(x,0) = u₀(x) satisfying u₀(0) = u₋(0), u₀(1) = u₊(1), where u± are given constants, u₋ ≠ u₊ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When u₋ 〈 u₊, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u₋ > u₊, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, that is, |u₋−u₊| is small. Moreover, when u_±(t) ≡ u_±, t ≥ 0, exponential decay rates are both obtained.     

An algorithm for tracing the boundary curves of mushy regions in some degenerate diffusion problems

We present an algorithm for approximating the solution of the degenerate diffusion problem u_t = (?(u))_xx in (0,1) × R⁺ (with zero Dirichlet boundary conditions, and nonnegative initial datum u₀), where ?(u) = min {ku1} for some ? > 0. The algorithm also provides an approximation for the interface curves which represent the boundary of the Mushy Region ?? = {(x, t): ? (u(x, t)) = 1}. The convergence of the algorithm is proved.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Numerical solutions with a priori error bounds for coupled self-adjoint time dependent partial differential systems

This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) u_t)_t = Q(t)u_xx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), u_t(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oR^r×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied.     

Regularization of a non-characteristic Cauchy problem for the heat equation

The non-characteristic Cauchy problem for the heat equation u_xx(x,t) = u₁(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), u_x(0, t) = ψ(t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate.     

Diffusion approximation for the convection-diffusion equation with random drift

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ _t u _ɛ (t, x) = κΔ _x (t, x) + 1/ɛV(t/ɛ²,xɛ) ·∇ _x u _ɛ (t, x) with the initial condition u _ɛ(0,x) = u ₀(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R ^d is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u _ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation. Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001     

Global existence,asymptotic behaviour,and global non‐existence of solutions for damped non‐linear wave equations of Kirchhoff type in the whole space

We study the global existence, asymptotic behaviour, and global non‐existence (blow‐up) of solutions for the damped non‐linear wave equation of Kirchhoff type in the whole space: u_tt+u_t=(a+b∥∇u∥^2γ)Δu+∣u∣^αu in ℝ^N×ℝ₊ for a, b⩾0, a+b>0, γ⩾1, and α>0, with initial data u(x, 0)=u₀(x) and u_t(x, 0)=u₁(x). Copyright © 2000 John Wiley & Sons, Ltd.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. I

In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u _tt−a ² u _xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1537–1544, November, 1998.     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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Asymptotic Behaviour of Solutions to some Pseudoparabolic Equations

The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem u_t−Δu_t−νΔu−(b, ∇u)=∇⋅F(u), u(x, 0)=u₀(x), where ν>0 is a fixed constant, t⩾0, x∈ℝⁿ. First, we prove that if u is the solution to the linearized equation, i.e. with ∇⋅F(u)≡0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long-time behaviour of u is described by the heat kernel. Next, analogous results are established for the non-linear equation with some assumptions imposed on F, p, and the initial condition u₀. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

We consider the periodic boundary-value problem u _tt − u _xx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u ⁰(x, t) + ũ(x, t), where u ⁰(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005.     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data

Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation u_t = u_xx + p(t)u + φ, in [0,1] × (0,T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ∫₀¹k(x)u(x,t)dx = E(t), 0 ≤ t ≤ T, for known functions E(t), k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ∫₀^s(t) u(x,t)dx = E(t), 0 < t ≤ T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems