Abstract
We consider an inhomogeneous semilinear heat equation with combined power and gradient nonlinearities with Dirichlet boundary conditions in exterior domains and half-spaces. Our objective is to determine the critical exponents governing the existence and nonexistence of global solutions.For the exterior domain, we establish Fujita-type nonexistence results in both the subcritical and critical regimes. The proof is based on the test function method combined with a harmonic weight. We further prove the existence of global positive classical solutions in the supercritical range for sufficiently small initial data.For the half-space, we introduce weighted test functions compatible with the boundary and derive nonexistence results in the subcritical regime. The critical and supercritical half-space cases remain open.