In this paper, the authors study the existence and uniqueness of bounded solutions of the Cauchy problem \[ \rho\partial_{t}u=L[G(u)]\text{ in }\mathbb R ^{N}\times(0,T]=:S_{T},\quad u=u_0\text{ in }\mathbb R ^{N}\times\{0\},\tag{1} \] where \(L\) is a second-order elliptic operator, formally defined by \[ Lv:=\mathrm{div}\{A(x,t)\nabla v\}+\langle b(x,t),\nabla v\rangle. \] Concerning the density \(\rho=\rho(x,t)\), the initial condition \(u_0\), the function \(G\) and the elliptic operator \(L\), the following hypotheses are made:

(i)

\(\rho\in C_{x,t}^{0,1}(\mathbb R ^{N}\times[0,T])\), \(\rho>0\);

(ii)

\(G\in C^1(\mathbb R )\), \(G(0)=0\), \(G'(s)>0\) for any \(s\in \mathbb R \setminus\{0\}\) \(G'\) is decreasing in \((-\delta,0)\) and increasing in \((0,\delta)\) if \(G'(0)=0\) for some \(\delta>0\);

(iii)

\(u_0\in L^{\infty}(\mathbb R ^{N})\cap C(\mathbb R ^{N})\);

(iv)

\(A\equiv(a_{ij})=(a_{ji})\), \(a_{ij}\in C_{x,t}^{1,0}(\mathbb R ^{N}\times[0,T])\) \(i,j=1,\dots,N\); \(\langle A(x,t)\xi,\xi\rangle>0\) for any \((x,t)\in \mathbb R ^{N}\times[0,T]\), \(\xi\in \mathbb R ^{N}\setminus\{0\}\);

(v)

\(b\equiv (b_{i}),\, b_{i}\in C_{x,t}^{1,0}(\mathbb R ^{N}\times[0,T])\), \(i=1,\dots,N\).

The authors formulate the assumptions on the coefficients of (1) in the form of the existence of the supersolution \(V(x)\) to the equation \[ LV : =\mathrm{div}\{A(x,t)\nabla V(x)\}+\langle b(x,t),\nabla V(x)\rangle = -\rho(x,t),\quad x\in \mathbb R ^{N}\setminus B_R,\;t\in (0,T],\tag{2} \] for some \(R \geq 0\).
To prove the existence of the solution to problem (1) satisfying a Dirichlet condition at infinity (\(\lim_{| x| \to\infty}u(x,t)=a(t)\) uniformly for \(t\in [0,T]\)), the authors assume that a suitable supersolution \(V=V(x)\) of (2) exists. More precisely, there exists a function \(V(x)\), which a supersolution to (2) in \([\mathbb R ^{N}\setminus B_R]\times(0,T]\), for some \(R >0\), such that

(i)

\(V>0\) in \(\mathbb R ^{N}\setminus B_R\);

(ii)

\(\lim_{| x| \to\infty}V(x)=0\).