Integral representation of the Cauchy problem solution for the heat equation

Question

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?

Answer 1

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 $\int_1^\infty dr/h(r)=\infty$.

Answer 2

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.