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The Fourier sine basis of the unit square is $e_{j\ell }(x,y)=2\sin (j\pi x)\sin (\ell \pi y)$, $j,\ell \ge 1$, 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 $\sum |a_{j\ell }{|}^2=\Vert u_0{\Vert }_2^2$. The solution isFor $\kappa \le 2{\pi }^2$, every exponent is nonpositive. The series converges in $L^2$ for $t\ge 0$, tends to $u_0$ in $L^2$ as $t\downarrow 0$ 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 satisfyExistence, 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 $|e_{j\ell }|\le 2$ and the Cauchy-Schwarz inequality:For $t\ge 1$, factor out $e^{2(\kappa -2{\pi }^2)t}$; the remaining sum is bounded by ${\sum }_{j,\ell \ge 1}{e}^{-2{\pi }^2(j^2+{\ell }^2-2)}<\infty$. ThereforeAt 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.