= Solution
The <Fourier sine basis> of the unit square is $e_{j\ell}(x,y)=2\sin(j\pi x)\sin(\ell\pi y)$, $j,\ell\geq1$, normalized in the <L2 inner product>. It satisfies the zero <Dirichlet boundary conditions> and $\Delta e_{j\ell}=-\pi^2(j^2+\ell^2)e_{j\ell}$. Expand the given initial datum as $u_0=\sum a_{j\ell}e_{j\ell}$, where $\sum|a_{j\ell}|^2=\|u_0\|_2^2$. The solution is
$$
u(x,y,t)=\sum_{j,\ell\geq1}a_{j\ell}e^{[\kappa-\pi^2(j^2+\ell^2)]t}e_{j\ell}(x,y).
$$
For $\kappa\leq2\pi^2$, every exponent is nonpositive. The series converges in $L^2$ for $t\geq0$, tends to $u_0$ in $L^2$ as $t\downarrow0$ by dominated convergence, and differentiates termwise for $t>0$ because the Gaussian high-frequency decay dominates every polynomial derivative factor. This constructs a classical positive-time solution and the appropriate mild solution at the possibly rough initial time.
Its norm and the difference of two such solutions satisfy
$$
\boxed{\|u(t)\|_2\leq e^{(\kappa-2\pi^2)t}\|u_0\|_2,\qquad\|u(t)-v(t)\|_2\leq e^{(\kappa-2\pi^2)t}\|u_0-v_0\|_2.}
$$
Existence, uniqueness in the natural $L^2$ solution class, and continuous dependence follow. For uniqueness one can project any weak solution onto each sine mode, obtaining the same scalar linear equation and initial coefficient. Thus this is a <Hadamard well-posed problem>, with a contractive <strongly continuous semigroup> in the stated range.
To prove pointwise decay from merely $L^2$ data, use $|e_{j\ell}|\leq2$ and the <Cauchy-Schwarz inequality>:
$$
\sup_{x,y}|u(x,y,t)|\leq2\|u_0\|_2\left(\sum_{j,\ell\geq1}e^{2[\kappa-\pi^2(j^2+\ell^2)]t}\right)^{1/2}.
$$
For $t\geq1$, factor out $e^{2(\kappa-2\pi^2)t}$; the remaining sum is bounded by $\sum_{j,\ell\geq1}e^{-2\pi^2(j^2+\ell^2-2)}<\infty$. Therefore
$$
\boxed{\kappa<2\pi^2\quad\Longrightarrow\quad\sup_{(x,y)\in[0,1]^2}|u(x,y,t)|\longrightarrow0.}
$$
At equality the first sine mode is stationary, so strict inequality is essential for decay of every datum. This is the <Dirichlet heat-reaction threshold on the unit square>. The equation is actually well posed on every fixed finite time interval for any fixed real $\kappa$; beyond the threshold it has a growing first mode and loses long-time contractivity. The restricted well-posedness assertion in the question is therefore sufficient rather than necessary.
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