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?



Integral representation of the Cauchy problem solution for the heat equation
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 $\int\_1^\infty dr/h(r)=\infty$. 


