For a class of non-linear stochastic heat equations driven by α-stable
white noises for α∈(1,2) with Lipschitz coefficients, we first show
the existence and pathwise uniqueness of Lp-valued c\`{a}dl\`{a}g solutions
to such a equation for p∈(α,2] by considering a sequence of
approximating stochastic heat equations driven by truncated
α-stable white noises obtained by removing the big jumps from the
original α-stable white noises.
If the α-stable white noise is
spectrally one-sided, under additional monotonicity assumption on noise
coefficients, we prove a comparison theorem on the L2-valued c\`{a}dl\`{a}g
solutions of such a equation. As a consequence, the non-negativity of the
L2-valued c\`{a}dl\`{a}g solution is established for the above stochastic
heat equation with non-negative initial function