AbstractWe study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$
λ
β
e
β
u
, forced by an additive space-time white noise. (i) We first study SNLH for general $$\lambda \in {\mathbb {R}}$$
λ
∈
R
. By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{8 \pi }{3 + 2 \sqrt{2}} \simeq 1.37 \pi $$
0
<
β
2
<
8
π
3
+
2
2
≃
1.37
π
. Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case $$\lambda >0$$
λ
>
0
, we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range: $$0< \beta ^2 < 4\pi $$
0
<
β
2
<
4
π
. (iii) As for SdNLW in the defocusing case $$\lambda > 0$$
λ
>
0
, we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical $$\Phi ^4_3$$
Φ
3
4
-model) and prove local well-posedness of SdNLW for the range: $$0< \beta ^2 < \frac{32 - 16\sqrt{3}}{5}\pi \simeq 0.86\pi $$
0
<
β
2
<
32
-
16
3
5
π
≃
0.86
π
. This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When $$\lambda > 0$$
λ
>
0
, these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on $$\beta $$
β
as in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for general $$\lambda \in {\mathbb {R}}$$
λ
∈
R
without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{4}{3} \pi \simeq 1.33 \pi $$
0
<
β
2
<
4
3
π
≃
1.33
π
, slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.
Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity