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?