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Consider the Cauchy problem for the heat equation $u_t=\Delta u$, $u|_{t=0}=\varphi$. S. Täcklind showed its solution $u$ is unique in the class $|u|\le e^{|x|h(|x|)}$, $|x|>1$, iff $\int_1^\infty \frac1{h(y)}\,dy=\infty$.

From the other hand if $|\varphi(x)|\le e^{cx^2}$ then at least for small enough $t>0$ the solution can be represented as a convolution with the the fundamental solution of the heat equation: $$ u(x,t)=\int_{\mathbb R^n}Z(x-y,t)\varphi(y)\,dy. $$

Why is there a difference between the uniqueness class and the class of initial functions allowing the integral representation?

To be more concrete, what can be said about the solution of the Cauchy problem with say $\varphi(x)=e^{x^2\log(1+x^2)}\sin x$? Its growth satisfy the Täcklind condition. Is there a solution from the Täcklind uniqueness class (not representable as a convolution of $\varphi$ with $Z$)? Or existence of such a solution is not guaranteed?

Are there analogous situations for other PDE?

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2 Answers 2

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In the book of Eidelman and Zhitarasu, Parabolic boundary value problems, Springer 1998, it is said that:

``using Tacklind's arguments in [76], it has been proved that if the initial function $\psi(x)$ satisfies the inequality $\psi(x)\geq \exp(|x|h(|x|))$, where $h$ is such that $\lim_{r\to\infty} h(r)/r = \infty$, then there is no strip $[0,T)\times\mathbb{R}$ in which the Cauchy problem for the heat equation admits a solution. A natural domain in which the solution of the Cauchy problem does exist is defined in this case in a special way, depending on the function $\psi$ and, generally speaking, is not a strip.''

The reference [76] is a paper by Eidelman and Petrushko at Ukrainian Math Jour 19 (1967), 93-97.

The non-existence of solutions in a strip, is restricted to the class of $u(t,x)$ such that $|u(t,x)|\leq \exp[|x|h(|x|)]$, if $\int_1^\infty dr/h(r)=\infty$.

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I suggest looking at the maximum principles contained in http://rspa.royalsocietypublishing.org/content/470/2167/20140079 and the references therein, which considers uniqueness results for solutions to linear parabolic inequalities, where the solution, and coeffients in the inequalities satisfy various growth conditions as $|x|\to\infty$.

Regarding existence, see Unconditional nonexistence for the heat equation with rapidly growing data?.

I am under the impression that the representation of a solution to the heat equation, as described above, and whether or not a solution to the heat equations is unique or not, are two fundamentally different questions.

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    $\begingroup$ Thanks for the reference, it mentions lots of interesting results on uniqueness. But the question here is not uniqueness by itself but why the Täcklind class is just slightly wider than the fundamental solution growth order? And does there exist a solution from the this class for the initial function $\varphi$ specified? $\endgroup$
    – Andrew
    Mar 31, 2016 at 17:10
  • $\begingroup$ So, regarding existence of solutions to the heat equation, with any smooth enough initial data (without growth restrictions), the link regarding existence (which guarantees that there exists a solution to the heat equation with initial data \psi (x) = e^{x^2\log{(1+x^2)}}$ adresses your query, with regard to the standard fundamental solution to the heat equation. Perhaps it is worth considering if there are alternative "non-standard" fundamental solutions to the heat equation, and what properties these might satisfy (or if a uniqueness result holds). $\endgroup$
    – JCM
    Apr 4, 2016 at 10:45
  • $\begingroup$ The question is: does any of such solutions belong to the Täcklind class? Nothing is said about that in the link. $\endgroup$
    – Andrew
    Apr 4, 2016 at 14:07
  • $\begingroup$ You are correct that the link does not necessarily consider solutions in the T\"ackland class, but it may be worthwhile to consider how the argument could (potentially) be adapted to allow the solutions to be in the T\"ackland class (since the argument does not simplt rely on integratinf the convolution of a fundamental solutions, and some initial data). $\endgroup$
    – JCM
    Apr 5, 2016 at 12:33

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