Determination of the Unknown Cauchy Data in a Linear Parabolic Problem from the Measured Data at the Final Time

The problem of determining the initial value u(x, 0) = μ ₀(x) in the parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the final overdetermination μ _T (x) = u(x, T) is formulated. It is proved that the Fréchet derivative of the cost functional ${{J(\mu_0) = \|\mu_T(x) - u(x, T)\|_0^2}}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. The existence of a quasisolution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method.     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

We are concerned with Friedrichs's scheme for an initial value problem u_t(t, x) = A(t, x)u_x(t, x), u(0, x) = u₀(x), where u₀(x) belongs to L_∞, not to L₂. We show that Friedrichs's scheme is stable in the maximum norm ·_L∞, provided that the system is regularly hyperbolic and that the eigenvalues d_i(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λd_i(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

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.     

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.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

Determination of the unknown source term in a linear parabolic problem from the measured data at the final time

The problem of determining the source term F(x, t) in the linear parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the measured data at the final time u(x, T) = µ(x) is formulated. It is proved that the Fréchet derivative of the cost functional J(F) = ‖µ _T (x) ? u(x, T)‖ ₀ ² can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

A direct numerical procedure for the Cauchy problem for the heat equation

For the Cauchy problem, u_t = u_xx, 0 < x < 1, 0 < t ? T, u(0, t) = f(t), 0 < t ? T, u_x(0, t) = g(t), 0 < t ? T, a direct numerical procedure involving the elementary solution of υ_t = υ_xx, 0 < x, 0 < t ? T, υ_x(0, t) = g(t), 0 < t ? T, υ(x, 0) = 0, 0 < x and a Taylor's series computed from f(t) ? υ(0, t) is studied. Continuous dependence better than any power of logarithmic is obtained. Some numerical results are presented.     

Locally Lipschitz solutions of a single conservation law in several space variables

We consider the Cauchy problem for a single conservation law in several space variables. Letting u(x, t) denote the solution with initial data u₀, we state necessary and sufficient conditions on u₀ so that u(x, t) is locally Lipschitz continuous in the half space {t > 0}. These conditions allow for the preservation of smoothness of u₀ as well as for the smooth resolution of discontinuities in u₀. One consequence of our result is that u(x, t) cannot be locally Lipschitz unless u₀ has locally bounded variation. Another is that solutions which are bounded and locally Lipschitz continuous in {t > 0} automatically have boundary values u₀ at t = 0 in the sense that u(·, t) → u₀ in L_loc¹. Finally, we give an elementary proof that locally Lipschitz solutions satisfy Kruzkov's uniqueness condition.     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I

We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived.     

Analytic-numerical solutions with a priori error bounds for a class of strongly coupled mixed partial differential systems

This paper deals with the construction of analytic-numerical solutions with a priori error bounds for systems of the type u_t = Au_xx, u(0,t) + u_x(0,t) = 0, Bu(1,t) + Cu_x(1,t) = 0, 0 < x < 1, t > 0, u(x,0) = f(x). Here A, B, C are matrices for which no diagonalizable hypothesis is assumed. First an exact series solution is obtained after solving appropriate vector Sturm-Liouville-type problems. Given an admissible error ε and a bounded subdomain D, after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy ε, uniformly in D.     

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u ₀(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to ? ∞. On the one hand, we give some conditions on u ₀ which guarantee that, for some λ > 0, the quantity c _λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim  _t→+∞ u(t, ct) = 1 (the stable steady state) if c < c _λ and lim  _t→+∞ u(t, ct) = 0 if c > c _λ. These conditions are fulfilled in particular when u ₀(x) e ^λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u ₀ for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ? and for all speeds c belonging to a given interval (γ₁, γ₂) with large enough γ₁ < γ₂ ≤ + ∞.     

Envelope viscosity solutions of first- and second-order PDEs with u-dependence

General envelope methods are introduced which may be used to embed equations with u-dependence into equations without solution dependence. Furthermore, these methods present a rigorous way to consider so-called nodal solutions. That is, if w(t,x,z) is the viscosity solution of some pde, the nodal solution of an associated pde is a function u(t,x) so that w(t,x,u(t,x)) = 0. Examples are given to first- and second-order pdes arising in optimal control, differential games, minimal time problems, scalar conservation laws, geometric-type equations, and forward backward stochastic control.     

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

On the Evaluation of the Tutte Polynomial at the Points (1, –1) and (2, –1)

Motivated by the identity t (K _n+2; 1, –1) = t (K _n ; 2, –1), where t(G; x, y) is the Tutte polynomial of a graph G, we search for graphs G having the property that there is a pair of vertices u, v such that t(G; 1, –1) = t(G – {u, v}; 2, –1). Our main result gives a sufficient condition for a graph to have this property; moreover, it describes the graphs for which the property still holds when each vertex is replaced by a clique or a coclique of arbitrary order. In particular, we show that the property holds for the class of threshold graphs, a well-studied class of perfect graphs.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations

Every solution u = u(x, t) of the Cauchy–Dirichlet problem for the fast diffusion equation, ? _t (|u| ^m-2 u) = Δu in Ω × (0, ∞) with a smooth bounded domain Ω of ${\mathbb{R}^N}$ and 2 < m < 2* : = 2N/(N ? 2)₊, vanishes in finite time at a power rate. This paper is concerned with asymptotic profiles of sign-changing solutions and a stability analysis of the profiles. Our method of proof relies on a detailed analysis of a dynamical system on some surface in the usual energy space as well as energy method and variational method.