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&x\\in\\mathbb{R}^n,& t>0,\\\\ u(x,0)&=u_{0}(x),\\ &x\\in\\mathbb{R}^n,&\n  \\end{array}\\right. \\end{align*} with prototypical external signal production \\begin{align*} f(x):=\\begin{cases} f_0\\vert x\\vert^{-\\alpha},&\\text{ if }\\vert x\\vert \\leq R-\\rho,\\\\ 0,&\\text{ if } \\vert x\\vert\\geq R+\\rho,\\\\ \\end{cases} \\end{align*} for $R\\in(0,1)$ and 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