Counter Examples in Differential Equations and Related Topics

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• A non-linear function u = u(x), continuous only in an open interval, which at every point x of the interval satisfies the one-dimensional mean value equation

• A non-linear discontinuous function u = u(x) satisfying the one-dimensional mean value equation (*) for arbitrary x and h = h(x) > 0.

• A discontinuous linear function

The following sections are included:

• A Differential Equation with no power series solution.

The following sections are included:

• An unbounded function b = b(x) for which both one-dimensional weak and strong maximum principles fail to hold in general.

• A positive function c = c(x) for which both extremum principles (Theorem above) fail to hold in general.

• An unbounded domain D for which both extremum principles (Theorem above) fail to hold in general.

The following sections are included:

• An Initial Value Problem with no solution.

• An Initial Value problem with local and no global solution

• A Line on which Initial Value Problem fails to have a solution.

• A differential equation with no real valued solutions.

• A differential system with no solution.

• An Initial Value Problem with no Lipschitz Condition.

• A general differential system with no closed path.

• An Initial Value Problem with no solution on any interval [0,ε), ε > 0.

• A general Cauchy Problem for hyperbolic equations with no solution.

• A general Cauchy Problem for elliptic equations with no solution.

• A nonlinear Cauchy Problem with no solution in the large.

• A general Dirichlet Problem for hyperbolic equations with no solution.

• A Cauchy Problem for Laplace equation : Δu ≡ u_xx + u_yy = 0 and data : u(0,y) = 0, u_x(0,y) = |y|, x > 0 with no solution.

• A Smooth Linear Partial Differential Equation in three independent variables without solution.

• A Partial Differential Equation failing to be locally solvable.

• The Initial Value Problem for the heat equation: u_xx = u_t with initial condition: u(x,0) = f(x) has no solution for t < 0 unless f is analytic.

• Domains (in three or more dimensions) for which an Elliptic boundary value problem has no solution in the strong sense.

• A Dirichlet Problem for Laplace equation: Δu ≡ u_xx + u_yy = 0 in domain G : 0 < x² + y² < 1 and boundary values:

• A divergent Dirichlet integral of the solution of a Dirichlet problem.

• A solution of u_xy = 0 in D(=ℝ²) is not solution of u_yx = 0 in D.

• A discontinuous function satisfying a partial differential equation without being a solution (in C¹(ℝ²)).

• “Never visit a universe where the temperature increases as exp(x²) towards the boundary”.

• An Abstract Cauchy Problem for heat-diffusion equation without solution in finite time.

• An Initial-Boundary Value Problem for the heat equation in multi-dimensional regions with nonlinear boundary data without global smooth solutions.

The following sections are included:

• An Initial Value Problem with at least two solutions

• The Differential Equation: with initial conditions: x(0) = 0 , x′(0) = 1 with two solutions.

• An Initial Value Problem: x′ = f(t, x), x(r) = ξ with convergence of successive approximations insufficient for the uniqueness of solution.

• A region in the xy-plane in which the Initial Value Problem: has more than one solution.

• An Initial Value Problem related to motion of an object dropped in vacuum from an altitude x₀, with two solutions.

• A Function U = U(t, u) Continuous on the (t, u)-plane such that the Initial Value Problem: u′ = U(t, u), u(t₀) = u₀ has more than one solution on every interval [t₀, t₀ + ε] and [t₀ - ε, t₀] for arbitrary ε > 0 and for every choice of initial point (t₀, u₀).

• A curve of the linear PDE of order two in two variables: au_xx + 2bu_xy + cu_yy + du_x + eu_y + fu = g, that is exceptional in the sense that the values of u, u_x, and u_y along the curve, together with the PDE, do not uniquely determine the values of u_xx , u_xy, and u_yy along the curve.

• The Initial Value Problem:
  admits two weak solutions.

• The Exterior Dirichlet Problem: Δu = 0 in Ω = {x : r = |x| > 1}, with u = 1 on S = δΩ: boundary of Ω, where neither |u| ≤ A in Ω, nor u{x) → 0 uniformly as |x| → ∞, has infinitely many solutions.

• A System of Partial Differential Equations of first order for which elimination and partial differentiation do not lead to an equation of second order, or one equation of third order.

• Solutions of Conservation Laws are not determined uniquely by their initial data.

• The Solution of Equation : u_xx = f(u) is not unique on the interval : 0 ≤ x ≤ 2π, where f(u) is a bounded function : f{u) = -u , -1 ≤ u ≤ 1.

• A homogeneous Dirichlet problem for an elliptic system of Partial Differential equations with infinitely many solutions.

• The Problem: u_xx - u_tt = 0, u(x, t) = f₁(x), on line x - t = 0, with , has infinitely many regular solutions u.

• The Problem : u_xx – u_tt = 0, u(x, 0) = φ(x) for 0 ≤ x < ∞, u(0, t) =ψ(t) for 0 ≤ t < ∞ in the first quadrant of the xt-plane, with φ(0) = ψ(0) and φ"(0) = ψ"(0) has infinitely many regular solutions u.

• The Problem: u_xx - u_tt = 0, u(0,t) = φ(t) and u(x, x/2) = ψ(x) for t ≥ 0 , x ≥ 0 in the angle-bounded domain D by the straight lines x = 0, t = x/2, t ≥ 0, x ≥ 0, with φ;(0) = ψ(0) and φ"(0) = ψ"(0) has infinitely many solutions u.

• The Problem: u_xx - u_tt = 0, u(x,x) = φ(x) for 0 ≤ x ≤ ∞, (u_x + u_t)|_t=x = ψ(x) for 0 ≤ x < ∞, with φ'(x) = ψ(x) has infinitely many solutions u.

• The Problem: u_xx - u_tt = 0, u(a,t) = u(b,t) = 0 for t > 0 in the half-strip a < x < b, t > 0 has infinitely many solutions u.

• The Cauchy problem : y²u_xx + yu_yy + ½u_y = 0 with data u(x, 0) = φ(x), u_y(x, 0) = 0 for 0 < x < 1, y < 0 has infinitely many solutions u.

• The Problem : with u(x,0) = φ₁(x), u_y(x, 0) = φ₂(x), u_yy(x,0) = φ₃(x) and condition , has infinitely many solutions u.

• A Nonlinear Example of an Elliptic equation for which the uniqueness fails.

• Non-uniqueness of the Cauchy heat problem.

The following sections are included:

• An unstable nonlinear plane autonomous first-order system which is asymptotically attractive.

• Pathological behavior of a system with a quasi-asymptotically stable solution.

• A nonlinear equation with unstable the zero solution and stable the linear approximation.

• A quasi-linear system with unstable equilibrium and asymptotically stable linear approximation.

• The speculation that the origin would be asymptotically stable for the equation: if there existed a Lyapunov function V with derivative DV negative definite is not true, in general.

• A Non-linear two-dimensional real autonomous system (NL) and the corresponding linear one (L) so that a node for (L) doesn't go into the node for (NL).

• A Non-linear two-dimensional real autonomous system (NL) and the corresponding linear one (L) so that a (proper) spiral point for (L) fails to go into a (proper) spiral point for (NL).

• A Non-linear two-dimensional real autonomous system (NL) and the corresponding linear one (L) so that a center for (L) goes to a spiral point for (NL).

• A Dynamical System with positively unstable motions according to Lagrange although the set Ω_p of all ω-limit points is ≠ 0.

• A Dynamical system with the disconnected set Ω_p (of all ω-limit points) in a non-compact metric space R, and with positively unstable Lagrange motions.

• A Dynamical System with a family of solutions stable in the sense of Poincaré and Lagrange, but unstable in the sense of Lyapunov.

• The solutions of the equation of pendulum: are, at exept for the origin, unstable in Lyapunov sense, but using new coordinates are Lyapunov stable.

• A Dynamical System with motions stable according to Poisson (including the rest points) and with motions unstable according to Poisson in both directions.

• An ill-conditioned initial-value problem

• The Cauchy problem for Laplace equation

• The Cauchy problem

The following sections are included:

• A Differential Equation with singular locus containing no singular solutions.

• A class of differential equations of 2nd order with no singular solutions.

• A differential equation of 1st order with an envelope of the family of solutions which is not a singular solution

• A differential equation of the 1st order with a singular solution which is not an envelope of the remaining solutions.

• A homogeneous linear differential equation of 2nd order with no solutions of the form (except y = 0).

• Implicit Function Theorem

The following sections are included:

• A Dynamical System whose integral curves recede in both directions but whose families of trajectories are , nevertheless, not regular.

• Nonperiodic solutions stable in the sense of Poisson

• A Dynamical System With Non-uniform Motion

• It is not always possible to orient a field of elements in the plane.

The following sections are included:

• The integral equation: possesses infinitely many linearly independent solutions for .

• A circle: |λ| < R, in which the eigenvalues λ of the integral equation: y(P) = λ ∫ K(P,Q)y(Q)dQ + f(P) cannot have a limit.

• A class of the integral equations with no eigenvalues.

The following sections are included:

• A real nonsingular matrix with no real logarithm.

• A differential equation with no single-valued solutions.

• A non-diferentiable subharmonic function

• An Initial Value Problem with continuity of first derivatives of initial data not yielding continuity of the first derivatives of the solution.

• A Cauchy sequence in the normed space of the continuous complex-valued functions f under norm failing to converge in this space.

• A functional J[f(x)] attaining a weak minimum on the curve f₀ = f₀(x) and not a strong minimum.

The following sections are included:

• Improperly posed problems of great potential importance in numerical analysis so far have not been reached by the main stream of active research.

• The existence problem for a regular transonic flow around given general profiles with given velocity at oo is difficult to be solved completely.

• The difficulty of the correct statement of the problems for equations of mixed type in higher dimensions still remains.

• It would be interesting to clear up the questions whether there is an extremal principle for the boundary value problems of mixed type.

• To consider regions of mixed type (elliptic-parabolic-hyperbolic) multi-connected with arbitrary parabolic curves is difficult to handle.

• The study of higher order equations and systems of mixed type equations is a question of great interest despite the little mathematical progress.

The following sections are included:

• REFERENCES

• SUBJECT INDEX

• NAME INDEX