Examples Of Non-uniqueness Research Articles

Some Remarks on Uniqueness for a Class of Singular Abstract Cauchy Problems

Of concern is the Cauchy problem for equations of the form $u''(t) + \alpha (t)u’(t) + {S^2}u(t) = 0(’ = d/dt)$ on a complex Hilbert space $X$. $S$ is a selfadjoint operator on $X$ while $\alpha$ is a continuous function on $(0,\infty )$ which can be unbounded at $t = 0$. Uniqueness results are obtained for these equations by applying a uniqueness theorem for nonlinear equations. Furthermore, nonuniqueness examples for the linear abstract Euler-Poisson-Darboux equation, which is contained in this class, show that the uniqueness theorem is best possible.

Some remarks on uniqueness for a class of singular abstract Cauchy problems

Of concern is the Cauchy problem for equations of the form u ( t ) + α ( t ) u ′ ( t ) + S 2 u ( t ) = 0 ( ′ = d / d t ) u(t) + \alpha (t)u’(t) + {S^2}u(t) = 0(’ = d/dt) on a complex Hilbert space X X . S S is a selfadjoint operator on X X while α \alpha is a continuous function on ( 0 , ∞ ) (0,\infty ) which can be unbounded at t = 0 t = 0 . Uniqueness results are obtained for these equations by applying a uniqueness theorem for nonlinear equations. Furthermore, nonuniqueness examples for the linear abstract Euler-Poisson-Darboux equation, which is contained in this class, show that the uniqueness theorem is best possible.

Non-uniqueness and prescribed energy for the continuity equation

In this note, we provide new non-uniqueness examples for the continuity equation by constructing infinitely many weak solutions with prescribed energy.